Well, I am against the death penalty for a number of reasons.
Let us start with the possibility of a framing, a mistake, or corrupt prosecution. It's too easy to make a mistake there and kill an innocent person. There are even enough instances of a couple having sex, the female getting angry the next day and telling police she was raped.
Next, let's move on to this: a person commits a crime that will result in execution. They now have no reason whatsoever to turn themselves in, or indeed to do anything except run and fight to the death. "In for a penny, in for a pound." and similar phrases actually resulted from times when people would be given the death penalty for a minor offence, and discover the freedom of "I can't make things worse."
Some time ago in China (as in, hundreds of years ago), a minister had a theory that if severe penalties were handed out for minor crimes, people would simply not commit crimes at all, and would never think of a major crime, because "If I could be harshly beaten for littering, or executed for skipping work for a day, imagine what murder would result in!" What happened, you ask? A few workers found themselves unable to work due to weather. It soon ended up with them realising "We are going to be put to death. You know, they're going to kill us no matter what, so let's at least fight this."
They ended up causing an armed revolt, and the minister was overthrown. This being ancient China, I imagine he was killed, and his immediate family exterminated, as well as his ancestors dug up and mutilated. Because they did that back then, to unpopular people.
So, a person murders someone now? Well, if they're going to get death for it, there isn't anything stopping them from loading up (let's tackle "No, you don't need to keep a selection of firearms in your house." another time, or not at all) and killing a few other people they don't like. Or perhaps any witnesses, the local police... basically anyone until they are finally taken down. They'll be more likely to take hostages, too, and to actually back up their threats and kill hostages if need be, simply because the worst that can happen is the default of what is already happening: they're going to die. They have nothing to lose, but something to gain.
Remember, laws don't exist to make you feel good about yourself. They don't exist to create justice in capital letters. They don't exist so that "people get what's coming to them." You want Abrahamic religions for that, not a legal system. The law is designed primarily for the same reason society is: so that people are rewarded for working with the system and being a part of it, and penalised for not doing so. In short, if you can fix vehicles and cook really well, but haven't the first clue about medicine, then even if you become self-sufficient for food, you're in trouble when you get sick unless you become part of the system. Be a member of society and you put your part in, doing a job and getting money for it, and then you get something out of it - you spend this money for things you can't do for yourself. And in some "communist" countries, they even look after you if you are sick or injured.
Likewise, if you work directly against the system, they make things difficult for you - you can't get things for free, and crimes have penalties attached to them. The legal system is designed to dissuade crime so that people work with the system as part of society.
Looking at it that way, you do want to always provide the option of "Stop what you're doing and cut your losses here." You can't do that the moment you decide to kill them. You can do that if they are looking at, for instance, a 20 year sentence (an actual 20 years as opposed to 2 years that might be increased up to 20), with it just getting worse for anything else they do. So they murder someone. 20 years. If they elect to try to escape the law, it's already getting worse, adding years onto it. Perhaps adding a fine as well, and taking privileges away in prison. They decide to take a hostage and barricade themselves in a building? Fine, it's just getting worse.
From this viewpoint, if you really have to kill someone, let's say you're a president with a hard-on for murdering people and you need to justify it, then the only crime that could warrant it would be the ultimate form of working against the system: armed rebellion/revolt. And you just don't see that these days.
Now, that being said? When I hear of a horrific crime, it sickens me and my first thought is that I hope they die painfully. But that subsides soon enough, because I know it's a base human reaction. It's violent and animalistic, but we all have it there. The trick is overcoming it.