There are 4 reasons why we might punish criminals. The death penalty serves 1, and arguably a second.
They are:
Retribution- This is pure justice. If the world were ending today, we would punish him any way.
The death penalty makes the most sense in this context.
Deterrence- We want to deter future crime. If the world were ending, we would not need to worry about future crime.
It is arguable whether there is a deterrent value to the death penalty.
Rehabilitation- We want to rehabilitate criminals.
The death penalty makes no sense in this context.
Incapacitation- To prevent criminals from continuing to do crime, we lock them up.
The death penalty does not incapacitate more than life in prison.
In Judaism, it is clear that we are at least also focused on retribution. Hence in parshas noach, the torah says, “shofech dam haodom, baodom damo yishafech”. I take that as retribution.
In my opinion, the principal reason why liberals and conservatives disagree on this is that liberals do not see society as justified in taking moral positions, which is a premise of retribution. Conservatives are more willing to allow society to make moral judgments, including that certain people deserve death.