I've been told that the laws had received widespread support amongst the Democrats of the time. This seems at odds with the current posture of the Democrat Party; is this true?
Yes. Following the American Civil War, the Democratic party was the primary haven for America's most machiavellian racists, (some of which set also included active criminals and terrorists), who labored tirelessly to subvert, frustrate, and stymie many of the postwar reforms, in a too often successful effort to continue a de facto slavery by other means and under other names.
Around the mid-20th century the Democrats grew to be somewhat less racist, leaving the more racist southern "Dixiecrats" without a party, which was when the Republican party resorted to what's been called the Southern Strategy to assimilate the Dixiecrats. Having done so, the Republican party has never been quite the same...
See also my answer to Are there cases of Democrats engaging in voter suppression?.
This is a classic case of being true on paper but needing extensive qualifiers to be properly understood by a modern day audience.
At the time when Jim Crow laws were introduced – 150 to 100 years ago, approximately – the US political arena was very different from what it is today. While the two main parties had been the Republican and Democratic party since the beginning of the Third Party System in the 1850’s, aside from the names of these parties many things have changed. In the 1860’s, the Democratic Party housed both Copperheads – anti-abolitionists who wanted to end the Civil War with a quick peace treaty and then negotiate the Confederacy back into the Union – and War Democrats – mostly abolitionists who wanted to restore the Union by defeating the secession states. Likewise, there were Radical Republicans – starch abolitionists and spearheads of Reconstruction – although it seems their counterparts within the Republican Party did not receive such a distinct label. (The Republican Party as a whole had been born from discontent on the expansion of slavery it had no strong anti-abolitionist fraction.) After the Civil War, the Republican Party was the dominant party winning all but two presidential elections.
Naturally, if one party dominates a country on a national level after winning a civil war which in turn was caused by part of said nation seceding to preserve the institution of slavery and disenfranchise the Black (i.e. slave) population, this winning party will follow through with their promises; accordingly, the Republican Party was the main driving force abolishing slavery and giving the Black population voting franchise and citizenship rights. Equally naturally, those who used to be in power in that area will strongly despise this new development and not support the winning party. Obviously however, the Republican Party had solid support among the freed Slaves. Furthermore, the freed Black slaves formed a majority in two states and a considerable voting block in many more. So those whites who had enjoyed dominance prior to the Civil War devised ways uphold their dominance.
Initially, this goal of keeping the freed slaves away from the voting booths was reached by violence and the threat thereof, see for example the Ku Klux Klan. Once majorities had been achieved in state legislatures, more subtle tactics for Black disenfranchisement were used which in and of themselves formed part of the Jim Crow laws.
The above mentioned developments were key points of the political history of the United States in the second half of the 19th century. At the turn of the century, the political issues shifted to what is now known as the Fourth Party System. The names of the parties were the same and the Republicans continued to dominate the northern and western states while the Democratic Party held the Solid South. The key change was personnel as the veterans of the Civil War were aging and dying while a new generation of politicians emerged. The major issues were now economic and labour issues, the results of the Gilded Age and the leadup to the Progressive Era. The major divide between Democratic and Republican fellowship was, however, religious with the Republicans enjoying the support of pietist Christians while the Democrats were supported mostly by Catholics and less pietist Protestants. In turn, this meant that there were supporters of both more and less economic regulation across both parties.
The Republican Party itself had two competing factions especially in the South: the Black-and-Tan faction which consisted of the Black members supporting equal rights and those whites (‘tanned’) who supported their cause, and the Lily-White faction which intended to exclude Blacks from party positions. The latter historically recruited most of its members from those who had supported abolitionism but did not want the Black-and-tan majority from gaining power within the party. Black disenfranchisement, although initiated by the Democratic party, helped sustain and strengthen the Lily-White faction. (All Black Members of Congress elected in the 19th century were Republican; the Democratic Party being seen as the party of secession, appeasement and slavery had very little support among the Black population.)
The next significant development occurred after the crash of 1929 and the Great Depression. This was, of course, the era of Franklin D. Roosevelt and his New Deal. His policies included social security and economic regulation, quite distinct from those of, e.g., Grover Cleveland, Democratic president of the late 19th century. The most direct effect of this was economically liberal Democrats being discontent with their party and tended to either fail reelection or cross the aisle to join the Republican Party. Thus, the 1930’s is when the Democratic and Republican Parties finally split somewhat cleanly on economic issues (Democrat Wilson, for example, had generally implemented similar policies to his Republican predecessors and successors). This is known as the Fifth Party System and it was the first time since the Civil War that the Democratic Party dominated the political landscape of the United States.
As FDR’s New Deal policies disproportionally supported the poor and as – following decades of slavery and then decades of disenfranchisement and segregation – Blacks where overwhelmingly poor, this also marked a period where the Democratic Party began receiving support from Black voters. However, both parties were still mixed or leaning conservative on social issues – especially the Democratic Members of Congress from the Solid South who benefitted the most from segregation and Black disenfranchisement. Indeed, conservative Republicans and conservative Democrats frequently worked together to prevent socially liberal legislation throughout the Fifth Party System era from the 1930’s to the 1960’s/70’s.
The first blow to this system may be considered President Truman’s Civil Rights initiative of 1948 which led to the Civil Rights Dispute at the 1948 Democratic National Convention. Truman said:
My forebears were Confederates … but my very stomach turned over when I had learned that Negro soldiers, just back from overseas, were being dumped out of Army trucks in Mississippi and beaten.
While this was fiercely criticised by the Southern Democrats who even walked out of the National Convention, fielded a third-party presidential candidate and carried four states, little happened legitatively with the aforementioned conservative coalition being able to prevent any compelling legislation from passing which was thus mostly not even introduced.
Anybody who studied American postwar history will know that at this point in time, the kettle was boiling and the Civil Rights Movement erupted in the late 1950’s and early 1960’s. While minor progresses were made during the Truman and Eisenhower (Republican) administrations, it was during Kennedy’s term that major legislation was devised and introduced – largely due to the immense pressure of the Civil Rights Movement, but it was Johnson who was able to get most of the legislation introduced in the late 1960’s. It is sometimes claimed but most likely untrue that Johnson foresaw the Solid South going Republican after these bills; what actually happened is that the Democratic Party as a whole and on a national level adopted a more socially liberal stance which alienated those southern conservative Democrats that had previously enjoyed hegemony in their area. This can be seen immediately in the presidential election of 1968, where among the states of the Solid South only Texas voted for the Democratic candidate Humphrey, while Wallace, a third-party candidate carried five (the presidency went to Nixon).
It’s worth taking a look at the Black vote over the ages which is somewhat related to the Black population distribution. Up until the 1910’s, nearly 90 % of the Black population still lived in mostly rural areas of the South where they had once been enslaved. These were often disenfranchised, turnout is reported to have been as low as 4 % in South Carolina’s 1st district in 1894. If Blacks were allowed to vote, they usually voted Republicans as they saw the Democratic Party being responsible for the situation they were in – although I have been unable to find numbers reaching this far back. Partial support to this claim can be found in the results of North Carlina’s 2nd district, known as the Black Second as it was gerrymandered to encompass most Black population centres, which almost consistently voted Republican between the Civil War and the 20th century.
During the First World War, a large number of able-bodied men were drafted to fight in Europe while the need for war material also boosted industrial economy – and industry was mostly located in the North. This led to a large migration movement of those willing to take up industrial jobs to these jobs, increasing the Black population of northern cities drastically (Detroit’s grew 2000 %, that is not a typo). In northern cities, the experience was no longer that one party strongly upheld segregation while the other had freed the slaves. Rather, the Republican Party had, over time, considered the white southerner vote more important than the Black southerner and equally noted that there was little ground to be made in the northern states by supporting equal rights. This reduced Black support for Republicans which in turn meant that at least some Black support for the Democratic Party must have developed. Sadly, I was unable to find numerical evidence.
Possibly the first big political event drawing Black voters to the Democratic Party was FDR’s New Deal – after the Hoover administration apparently drove them away from the Republican Party by his handling of the Mississippi flooding. As a nice graph from factcheck.org (reproduced below) shows, Black party identification was almost 50:50 split between Democratic and Republican at the beginning of the New Deal era in 1936 but Black party identification soared to almost entirely Democratic after the Second World War and the Civil Rights Act passing. Perhaps more indicative are the graphs that the Washington Post displays using the same data: the Black vote was about 70 % Democratic in the New Deal era, plunged to 60 % for Eisenhower’s re-election and then soared to consistently above 80 % after 1964. It’s worth comparing the Black Democratic vote to the overall Democratic vote: the two curves are almost parallel from 1936 to 1964 (although Truman dented the Black curve upwards, i.e. had more Black than non-Black support), but the Black Democratic vote almost completely separates itself from the overall Democratic vote post-1968.
In the decades since the 1960’s, a slow realignment within the parties has been taking place. It meant that conservative Democrats felt less at home and less happy with their party at a national level and thus heavily socially conservative areas would slowly realign to voting majorly Republican – usually once a Democratic incumbent retired. The 1960’s also ended the Black-and-Tan and Lily-White factions within the Republican party with the latter mostly taking over (although Black-and-Tan had been a minority or non-existent in most areas already). Taken together, many observers argue that the US are now in a Sixth Party System, in which the Democratic Party is socially and economically progressive (and strongly opposed to Jim Crow Laws) while the Republican Party is economically liberal and socially conservative. So to a present-day observer a mere answer of yes is not giving the full picture.
The above is an attempt at a brief overview. It leaves out a lot of details and probably gets others wrong although I did my best to be as accurate as possible.
If you want an even briefer overview, I can recommend this 24-minute Youtube Video which I feel explains it well while debunking a couple of common misstatements along the way.
The existing highly upvoted answers do a good job of explaining why the answer to this question is "yes, but significant additional context is needed." I add here as an addendum a brief and visually intuitive aid to add that context.
Third Party System
Below is a political map of the Third Party System in the U.S., which lasted roughly from 1856-1892. This followed the end of the civil war and collapse of the Whig party.
At this time, the Republican party platform supported national banks, higher social spending, and aid to land-grant colleges, while The Democrats supported laissez-faire capitalism and lower tariffs. Both parties also had strong opinions on issues less salient today, such as the gold standard and railroads. This is a fairly clear inversion of current party politics, which is where your question about Democrats supporting Jim Crow laws is answered. The Democrats base was the south, and the south supported such policies. We need to keep moving to gain context on the party switch.
Fourth Party System
Below is a map of the Fourth Party System, which lasted roughly from 1896-1928. For the purposes of answering your question there isn't much additional to note here
Fifth Party System
The Fifth Party System began with the election of FDR and was characterized by the New Deal - a slate of economic policies that appealed to most people at the time.
(The usual picture for this is missing from Wikipedia, I'll update it when it returns. Or someone else can. Gets the point across anyhow)
Many that were not Democrats before became Democrats, and the Republican party faced a fracture between conservative and liberal wings. While the GOP initially failed to coalesce under the conservative banner of Goldwater, they did so definitively under Reagan. The progressive economic policies of the Democrats came to define the party, but following the Civil Rights Act of 1964, the GOP employed the Southern Strategy to court southerners whose support of the economic policies was outweighed by their... social policies.
Sixth Party System
And that brings us to the Sixth Party System, which started at the end of the Fifth Party System (when this ended is disputed, some say 1964, others say 1980).
This certainly lasted until 2016, but as it seems that Blue Wall states like MI, WI, and PA are trending redder and Sun Belt states like AZ, NM, NV, and GA are trending bluer, I would posit that we are undergoing the transition to a new party system. Time will tell.
To Answer Your Question
Yes, the Jim Crow laws were championed by the Democratic party. This is because the southerners who desired such a system were, at the time, the base of the Democratic party.