Why do we kill people who kill people to show that killing people is wrong?
“At the heart of Catholic teaching on the death penalty is the belief that ‘Human life is sacred because from its beginning it involves the creative action of God and it remains forever in a special relationship with the Creator, who is its sole end…’ (Catechism, No. 2258).
Regarding the death penalty, the Catechism of the Catholic Church states:
Assuming that the guilty party’s identity and responsibility have been fully determined, the traditional teaching of the Church does not exclude recourse to the death penalty, if this is the only possible way of effectively defending human lives against the unjust aggressor.
If, however, non-lethal means are sufficient to defend and protect people’s safety from the aggressor, authority will limit itself to such means, as these are more in keeping with the concrete conditions of the common good and more in conformity with the dignity of the human person.
Today, in fact, as a consequence of the possibilities which the state has for effectively preventing crime, by rendering one who has committed an offense incapable of doing harm — without definitively taking away from him the possibility of redeeming himself — the cases in which the execution of the offender is an absolute necessity “are very rare, if not practically non-existent” (#2267).
-The above section is from “Questions and Answers” on the USCCB website about the Death Penalty/Capital Punishment.