It's hard to believe that the Democrats of today are more liberal than those of the civil rights and strong unions periods.
Actually, it's quite easy. First, you have to realize that in those periods, the modern state would be considered quite liberal.
On civil rights in particular, no one is talking about rolling back any of the old laws. At worst, the conservative Supreme Court is. No Democrat opposes the Voting Rights Act. Contrast that with the actual results in 1965 when seventy-one Democrats (out of two hundred sixty-six) voted against the Act while only twenty-one Republicans did (out of one hundred sixty-two). That's more than a quarter of the Democrats against just over an eighth of the Republicans.
Similarly, can you see any pro-labor forces asking for card check when unions were increasing? At the time, it was common for roving bands of thugs to beat up union and company sympathizers (because both company and union had their own thugs). Secret ballots was to prevent companies from retaliating against workers for voting for the union.
Unions are mostly victims of their success. They've made repetitive jobs so expensive that most are automated. Also, worker safety and wage requirements make unions less important. Why unionize for a minimum wage job? You can get the same job without paying dues. The laws are about as pro-union as they've ever been. Even if their enforcement is less so.
Remember that for years, it was the Republicans who were the liberal party while the Democrats were the conservatives. As late as 1932, FDR ran on a balanced budget plank against the reckless spendthrift Hoover. The New Deal? Those were mostly programs that Hoover proposed during the election. FDR rebranded them and got them passed.
It was when the South switched from the Democrats to the Republicans that this changed. Until then, Southern Democrats were routinely more conservative than New England Republicans. Now, even the most liberal Republican is more conservative than the most conservative Democrat.
Finally, realize that even if Congress is becoming more conservative, that the Democrats aren't. Notice that Congress became more liberal in 2006 and 2008 even though the Democrats became more conservative. This was because the balance shifted. A bunch of comparatively conservative (for modern Democrats) Democrats were elected. This shifted the party more conservative but shifted Congress to be more liberal.
As others have noted, the new Democratic issues would have been considered quite radical in the past. Gay marriage? Until the nineties, homosexual acts were still against the law. Government mandated birth control insurance? Prior to Griswold, birth control could be outlawed. Immigration? Remember during World War II when a Democratic Congress and White House rounded up Japanese people into camps? In its own way, that's as bad as Trump's Muslim ban.