Every once in a while, something awful happens. And then someone will say "the people who do that should get the death penalty". Invariably, that is a stupid argument.
Let's start by disposing of the obvious low blow: "if it happened to your son, you would ask for their death too". To which the obvious answer is, of course I would. And I would be pushing for a stupid solution to the problem.
If something really, really bad happened to my son, I would be suffering, and in a state of violent emotion and distress. If I told you to jump off a bridge, would you? I guess not, because you would notice I am at the edge of madness and just told you that because I am feeling that way.
Well, the same is true about a victim's relatives (or the victim himself) asking for a specific punishment: it's a request born from pain and suffering. And as a society, we should not decide our actions based on te proposals of the ones almost insane with grief. Because we want to take the action that is better for society, not for a specific member of it. So, forget about that appeal to sentiment, because while completely understandable, it is stupid, because those grieving are allowed to be stupid.
Another failed argument for the death penalty is that it discourages future crime. This has been shown not to be the case, because similar countries or states with and without death penalty show no significant difference on crime rates.
I suspect this is because in many cases the criminal doesn't intend to actually commit the part of the crime that brings the death sentence (murder when robbing? The criminal wanted to rob, not to murder) or he just would do it anyway (it's not as if child rapists expect to have a nice time when captured. They just do it assuming they won't get caught, or decide it's worth it anyway).
And of course, the main reason why the death penalty is a bad idea is that it would be applied by public employees. Do you trust the government to decide how much to charge you in taxes? No, you do it yourself, with an accountant, and tell them how much to take. Do you expect the police to find a wallet you dropped on the street? Do you trust the government's mail with really important packages? Are you confident about the government figuring out all environmental issues? Well, then why would you trust them to kill anyone they decide is a really, really bad guy?
The government is a great idea for things noone else wants to do (highways), or things noone else should be doing (law enforcement) but even in those cases you must allow for them providing a crappy service, and carefully use your input to limit what they can do.
The alternative is to assume that the government has a huge capability and competence but has just decided not to show it, which is perhaps the most amazing part of most conspiracy theories.
The US government can't kill Castro, but can kill Kennedy. A government can't keep the trains running, but can fake a moon landing. And so on, and so forth.
I once read a book where they described an interesting science: Conspiracy Theory Theory. It's the science of studying what conspiracy theories people believe in, to get insight into what they actually believe. If you are in favour of the death penalty, then you believe law enforcement doesn't make mistakes, or that killing a few random people every now and then (because they make mistakes) is not a problem. If the first, you are stupid, if the second, you are evil.