The only reason people believe we have a voter fraud problem is because some prominent people keep saying "we have a voter fraud problem" and many people believe them, even though it's not really true. Check out the Heritage Foundation's (right-leaning think tank) voter fraud database. There's basically none to speak of, and there are already systems in place to prevent it.
The link I posted suggests voter ID laws, specifically, have not suppressed turnout, hence why I have said I don't have a huge problem with that concept in principle, as long as said voter ID is free and easy to obtain. That does not mean that the intent isn't to suppress votes with voter ID laws, even if they are unsuccessful in doing so. From the article:
Longtime Republican consultant Carter Wrenn, a fixture in North Carolina politics, said the GOP’s voter fraud argument is nothing more than an excuse.
“Of course it’s political. Why else would you do it?” he said, explaining that Republicans, like any political party, want to protect their majority. While GOP lawmakers might have passed the law to suppress some voters, Wrenn said, that does not mean it was racist.
“Look, if African Americans voted overwhelmingly Republican, they would have kept early voting right where it was,” Wrenn said. “It wasn’t about discriminating against African Americans. They just ended up in the middle of it because they vote Democrat.”
Or from the founder of the Heritage Foundation:
Or as Trump himself put it, if there were more mail voting, “you’d never have a Republican elected in this country again.”
There is also no legitimate voter fraud argument for restricting early voting days, poll hours, or the registration window. Those are blatant attempts at voter suppression. Voting a week early, or at 9:00 PM, or on the same day you registered is no more susceptible to fraud than voting at noon on Election Day after registering months before.
Democrats would like to keep the "special" rules because, as we can clearly see, voter turnout was especially high this cycle with those accommodations in place. Democrats, like Republicans, are political creatures, and they know that higher turnout has traditionally increased their odds of winning. But, the fact that Democrats may benefit from it doesn't change the fact that more people voting is a good thing in a democratic country. Both parties are fighting for rules they think will help them win. It just so happens that Republicans believe getting fewer people to vote will help them win while Democrats believe getting more people to vote will help them win. In a vacuum, one of those two goals is more virtuous than the other.