I am in favor of requiring a picture ID. That would be one of the best impediments to voter fraud we could have, and the cost would be minimal.
Just to add one thing - no constitutional amendment would be needed. Do it on the State level. Federal election law could incorporate the same.
I think that, much like applying for a driver's license or a permit, you should be able to produce 2 positive forms of identification (voters selected at random) to vote. It would be a major deterrent to Mary Poppins, Darth Maul and Puff the magic dragon.