This debate is a classic.
As people arriving at an answer typically do so from two different paradigms, very often agreement between the two schools of thought is impossible; it's like the abortion debate.
To some people an unborn baby is a soul and is sacrosanct. The arguments secular people use are irrelevant.
To secular people religious predicated arguments are irrelevant.
Likewise with the death penalty.
Some people believe if killing is wrong, then killing is wrong.
Others believe you can have a good excuse to kill someone (other than self defence in the heat of the moment).
It is an even more curious debate as it is not just a stereotype to say that many pro-lifers favour the death penalty, and many pro-choicers are against it. The image of someone going from an abortion clinic picket to a pro-death penalty vigil outside a prison on an execution night is a strange one, but it probably has happened.
From a European perspective it's an obvious cultural issue.
Grow up in a religiously active violent culture where the death penalty is enforced and you will feel it is right.
Grow up in a secular culture with lower levels of violence and no executions and it will appear barbaric.
This is probable historically; in Germany the new government (after the war) banned the death penalty at a time when most Germans were still in favour of it. Two generations later Germans were one of the countries MOST opposed to the death penalty.
What I find extraordinary is that some Americans do not connect the dots.
There they are, one of the most execution happy governments in the world alongside those bastions of human rights such as China, Saudi Arabia and Iran. And it never strikes them as inappropriate for a modern democracy and an indication their continuing enchantment with judicial killing is out-of-step with the rest of the civilised world.
Of course, those very people this is directed at come to their conclusions using a different paradigm to me, and will disagree with me.
Arguing with a fervent supporter of the death penalty is like convincing a supporter of Sharia Law it's not a good idea to execute people; your arguments are meaningless to them, just as theirs are meaningless to you.