Constitutional FAQ Answer #73
Q73. "Where are the requirements for the right to vote spelled out? Specifically if you were dishonorable discharged from the military, are you ineligible to vote for president?"
A. It depends on your state — a state cannot deny the vote solely on your gender or race, etc., but it could have other requirements, one of which could be denial for dishonorably discharged veterans. I doubt it, though. More likely would be convicted felons currently in prison, and the like.
I did find an interesting tidbit: literacy tests are legal, as long as every voter has to take one. Logistically, it would not be possible, but it is legal (42 USC Sec 1971.a.2.C).