The primary way that the Democratic party suppresses the vote is by clumping. For example, in Massachusetts, there is roughly one Republican for every three Democrats by voter registration. Yet there are no Republican members of Congress from Massachusetts. If Republicans had as much representation as their share of the two-party registration, they would have two or three Representatives. Even if Republicans only had as much share as their overall percentage of registrations, that would be still be at least one. And of course, Democrats would be reduce from nine to three. Because unaffiliated voters make up a majority of the voters in Massachusetts.
This also has an effect because it makes elections uncompetitive in Democratic areas. Much has been made of California's huge lean towards Democrats. But again, Republicans get about a third of the vote but only a quarter of the seats. And in many elections, like Senate, governor, and president, California is completely noncompetitive. So there is little incentive for Republicans to vote. Lopsided elections reduce turnout.
It's true that these effects cut both ways. In places where Republicans clump, it's Democrats whose vote is suppressed. But the truth is that the most partisan areas are controlled by Democrats. And with Democrats, there are usually Democrat enclaves within the Republican areas. For example, Texas is a Republican state. But Houston, Dallas, and Austin all have Democrat mayors and Representatives. Fort Worth, San Antonio, and El Paso may not have Democratic mayors but still have Democratic Representatives.
Of the fourteen Representatives from New York City, only one is a Republican. Republicans who live in other districts don't have the same opportunity to cast votes that count.
Both Democrats and Republicans benefit from the two-party system, but the Democrats benefit more. The two-party system forces people to pick one side or the other. We can again see this by looking at Massachusetts. There, a majority of voters are unaffiliated. They should have four or five Representatives but actually have none. Democrats get all of the seats despite only making up a third of registrations.
Democrats also apportion and district seats by total population rather than voter eligible population. So someone who lives in a district with many children and disenfranchised felons has a vote that counts more than those in older, less criminal districts.
One could say that these are demographic problems. That's true. But they remain addressable. If we adopted proportionality, then seats would be distributed based on the number of people who vote. We could get rid of the census, redistricting, and taxpayer financed primaries altogether. There would just be one big election via proportional means (e.g. single transferable vote). Candidates could continue to be limited to states, but apportionment would happen as part of the election. That would incent all groups to participate.
Instead, Democrats came up with efficiency gap analysis, which suppresses third party votes, doesn't fix Massachusetts, California, and other partisan Democrat states, but does help Democrats in Republican states.
Proportional voting would make it easier for minorities and women to win as well. Because they could clump together to vote for their candidates. As is, Massachusetts is represented by seven white men and two white women. Under proportionality, that might be more even by gender with two or three minority Representatives.
Clumping combines with partisan primaries to increase polarization also. Because electability doesn't matter in a clumped district. Even an obviously unqualified candidate can expect to win the general election after winning the correct primary. And it's much easier to win a primary than a general election. Because primaries only include a partisan group of voters, they tend to produce candidates who are ideologically in the middle of that subset rather than the population as a whole. Meanwhile, if offered the choice, the voters excluded from the primary would prefer the more moderate candidate.