Give the FEC the power to issue photo ID for voting. Legislate that it is only valid for that purpose and any other use is illegal. Send out census-style teams to track down every citizen and issue them a card, free of charge.
Of course, there are all sorts of problems with this that are entirely self-inflicted. Someone will oppose a national ID, even if we already have that. Someone else will point out that the states legally get to make their own voting rules, and they're right. This isn't a problem anyone actually wants to solve. Democrats want to make it easier to vote, Republicans want to make it harder. It's like every other tribal issue, neither side will give an inch because they fear the other side will take a mile.