If you think ineligible voters are not a problem then you don't have any experience with it.
We have secondees who work for us from South Africa. One of them just recently got a voter registration card in the mail. He was freaked out because he knows its a felony to even register if your aren't eligible and didn't know what to do. Turns out when he was renewing his license there was someone who was standing outside the DMV that told him if he filled out a form with certain information it would help him move faster once inside. He distinctly remembered it not saying anything about registering to vote, and in the "Country" section he even put down South Africa. He assumes that was actually someone who was trying to register people and had trick forms to get their information and sign them up. He was able to call the Supervisor of Elections and get his card revoked, along with a letter so he can't be persecuted for it, but assuming he never reported himself he would have been able to go cast a vote.
I don't think the idea of making sure someone is eligible to vote is voter suppression in any way. If you are eligible to vote you can easily prove it. The only people this hurts are those who are illegally registered to vote.