What if two politicians are running for office and neither is qualified? Is there an ethical duty to vote for the lesser of two evils, even if doing so results in putting a stamp of approval on an unqualified candidate? Or is there an ethical duty not to vote for either of them, since doing so would give them legitimacy?

There is no ethical duty to vote or not to vote. Only a person who is an act utilitarian view holds that people have an ethical duty to do whatever has the best consequences, and this view seems to require far more than most people would accept. Especially in this case, when it is not even clear whether voting would have better consequences than not voting, it would seem quite extreme to claim that one had a duty to do either. However, if you think that one of these choices has better consequences than the other, then it would be a sign of a good moral character to act in the way that you thought would have the best consequences, or in this case, the least worst consequences. It was pointed out to me by Wm Derek Bowman, Philosophy Grad Student at Tulane University that my answer was incorrect. It is not only act utilitarians that might claim that you have a duty to vote. Almost any revisionist ethical theory can claim that you have a duty to vote. However, if we are using "duty" in the ordinary...