Christ died for that man’s *sin*, not crime. The crime is against society, and if that society has deemed execution to be the rightful result of committing a specific act, then the criminal has chosen to forfeit his life.
Christ did not absolve people of their responsibility to their society, but rather offered a personal mechanism for redemption for each soul.
When Christ was on the cross, he told the thief on the cross next to him “Tonight you will be with me in paradise.” (Sorry if I have the exact quote wrong.)
He did not save the man’s life, but rescued his soul, as his life was forfeit to the state as punishment for his crimes.
When faced with a capital criminal and challenged to condemn her to death, Jesus instead said, “I do not condemn you” (John 8:11).
Responsibility, consequences, and society’s appropriate response to crime are embodied in incarceration for the reasons I’ve mentioned.
The reason people believe in punishment by death is mainly because it seems like justice for the man’s acts. But full justice and the death penalty has already been dealt. So society must look elsewhere for appropriate consequences and rationales to deal with criminals. I have named a few of them.