No state has an absolute right to put its worst criminals to death although a majority of a state's residents may wish to confer that right on it. Of course all states do kill people, even where they do not have the death penalty. Our police are armed (by the state) and people get killed in shoot outs with them.
Majority opinion is typically in favor of the death penalty, with recent surveys indicating around a 60 Ð 65% level of support. Opponents believe that it is wrong for the state to kill, per se.