Forget about the fact that sometimes innocent people get executed. Forget about how expensive it is to execute someone. Forget about the idea that the state shouldn't have the power to kill its own citizens.
Even forgetting all of that, I still can't see why you'd want to execute someone.
Is it that people want justice? Revenge? (What's the difference, here?) If so, fine. Is killing somebody really justice? Would it not be more just for someone to rot in a cage for all their years, instead of receiving the mercy of death? You can't teach a lesson to a dead man.
What's the point?