Convicted criminals still have certain rights enumerated in the Constitution. Popular demand should never change that-this is a case where majority sentiment is inappropriate because it is by definition biased.
Your argument could be easily disproved by offering death-row convicts the option to have their sentence commuted to life-I would submit that most (although possibly not all) would take the offer.
Also, judges and juries are human and make mistakes. If that mistake is later uncovered, a living prisoner is helped by that-a dead one is not.