That largely seems to be an imagined epidemic, like 'anchor babies'. Illegal immigrants generally keep a low profile, they try to avoid any potentially illegal activity that might get them in trouble with the law. If you see a car driving exactly at 65 mph on California highways, a lot of the time, it'll be illegals, because why would they want to draw attention to themselves?
There is no evidence, as far as I can see, of massive numbers of illegals actually showing up to vote. There is, however, evidence that putting roadblocks into the voting process tends to discourage people. We already have tragically low participation rates in our democracy, so voting should be as streamlined as possible.
I have no problem with coming up with ways to verify that voters are indeed citizens. But it has to be efficient and fair. Most of the most economically disadvantaged voters don't necessarily even have stable addresses or drivers licenses.
Yes, I'm willing to bet that Democrats party registration efforts go into areas and get people to sign up for voter registration rolls who may not be eligible. It's like signing petitions, people have quotas and concentrate on boosting numbers.
But whether or not those people whom political parties help register actually show up to vote on election day is a different story. It doesn't surprise me that the GOP takes the lead on voter purges, because they are running against long run demographic challenges: racial and age composition of the US is changing and it does not favor the current GOP voter demographic.