Right, a bit of categorization would help us all out here methinks.
There are two aspects to the (il)legitimacy of capital punishment: the principle and the policy.
Examples of principle-based arguments against capital punishment would be (from the answers given by above posters): it doesn't solve anything, it's too barbaric, etc. These arguments are against the very concept of death penalty, regardless of how it is implemented.
Examples of policy-based arguments against capital punishment would be: it's difficult to ascertain the crime (if, on the other hand, someone argues that death penalty is unjustified because we can never be above reasonable doubt about murder under any circumstances, then it would become a principle-oriented argument), prison life for those to be punishable by death being very cruel, etc.
I'm more interested in the principle-orientedness, so will only talk about that.
I'm surprised no one brought up the "deterrent" argument (is it too politically incorrect? I have no idea). In its crudest sense, the dictum is that harsh punishments tend to scare off potential murderers, thus bringing the overall murder (or crime) rate down.
The possible defeaters I can think of:
1. It's a slippery slope. A valid worry, historically a lot of governments that adopted the deterrent argument did go overboard with it, killing people left and right. But that's a policy concern, more than a principle one. Secondly, if we can say that we are civilized enough to move on from "barbaric" types of punishment, why can't we say that we are civilized enough to move on from barbaric, over-the-top manipulations of punishments?
2. A civilized society wouldn't require such barbaric measures, since the social structure would be advanced enough to pre-emptively reduce crime rate. True, but that's not to say it cannot be complemented with a strong deterrent. In the case of such a society with a top-notch anti-crime social structure would have very low crime rate as is, which would render the capital punishment no more than a fear-factor. In fact, I can conceive of a society where capital punishment is rarely established, because the very concept of capital punishment is deterrent enough to lessen such crimes.
3. Statistics. I don't know of them, but perhaps people can cite cases where capital punishment failed to serve as a deterrent. In such cases, the penalty would seem utterly superfluous. I would argue such cases are failures at policy levels, not at principle levels. In terms of principle, capital punishment makes sense. So if the implementation is unsuccessful, something is probably wrong with the policy as opposed to the principle.
Argument for capital punishment.
Consider this: if you adopt a utilitarian concept of justice, then it would include the prospect of benefiting human persons or society. This serves as at least one of the premises for the deterrent argument: the punishment of one individual brings about the greater good of the society. I know "greater good" is a dangerous word, but I think our minds associate the danger of greater goods with it's slippery slope-ness. That possibility can be averted by adopting strict policies.
In the above discussion, I think I've been discounting the slippery slope argument way too fast. But I simply don't think such an appeal valid in the present case. I would love to be proved wrong, though.
Also, please don't address these by policy-based arguments. That would count as red herrings and I would ignore them.