If the organs are healthy, then death row inmates should donate their organs after death. That way their debt to society is truly paid in full. There are no reasons why someone who dies in a state execution should not donate. The government has owned a death row inmate for decades and the person already has limited rights isolated from society.
I believe if they are healthy and willing, yes, death row inmates should donate their organs. There are billions in the US alone in need of a necessary transplant to live, and organ donation is not for everyone. There is, however, and abundance of death row inmates who could help save lives, if they were willing to. This should be allowed to be an option given to them if they choose. A lot of lives could be saved.
I don't, however believe that they should be forced to do so if they specifically say 'no'. A persons body is precious to them even if they have done horrible things with it, this should be respected. But, I think it would be a good idea to ask suitable candidates if they would be willing to do so. Even if you're on death row it's quite possible that you may still be willing to do a last good deed before you die. With this in mind, I think that it should be given as an option before death, just like a choice in last meal or a last phone call. The results may surprise many people!
While I believe organ donation serves a great purpose and everyone should consider becoming one, I do not think we should tell anyone what to do with their body regardless of what they may have done in life. They may have religious beliefs or other reasons as to why organ donation would not be good for them so it should be their choice.
Why should they be forced to donate organs just cause of a crime committed? I don't think that is reason enough to remove their right to donate or not away from them. If they want to donate that's fine, but otherwise not. And what of their family? Shouldn't they have some say in the matter too? So no, I don't think death row inmates should donate their organs, it should be left up to them and to their immediate family.