The Amish and Mennonites, as a matter of religious conviction, also choose not to excercise their right to vote. Additionally, should those who refuse to be photographed also not be denied the right to drive and have a driver’s license? Surely you wouldn’t advocate they be denied the right to economic progress and independence that comes along with the right to drive.
Asking for ID when voting, just as it is required when entering many buildings, flying, purchasing alcohol or applying for food stamps or Medicaid, is a very minimal requirement. And it insures the integrity of the vote.
Given the general agreement that committing voter fraud is extremely difficult to detect since generally no one will usually realize when someone voted in someone else’s name, let alone the difficulty to identify the perpetrators for prosecution even if it is detected, requiring photo ID seems to be a very reasonable and minimal solution.