There are at least three different kinds of justification for intentionally punishing those who have done something wrong. Only one is utilitarian in nature and it is the grounds usually discussed in the media when issues such as the justification of capital punishment are raised. Yet it is less relevant to capital punishment than the other two. 

Of course, these justifications are about punishing the guilty, not about mistakenly punishing the innocent, even if they are thought to be guilty and even if, in a court of law, they are mistakenly convicted of a crime they did not actually commit. It also presupposes that the act for which one is being punished actually is something that is wrong to do, and is not something that is merely unreasonably illegal or unjustifiably and unreasonably regarded as wrong.

1) Punishment as a deterrent of future wrong-doing, where it works to deter either the same person's repeating the offense (or doing anything else wrong) or where it serves to deter others from committing the same (or any other) offense. Punishment sometimes is a deterrent, and as long as it is not a draconian or unfairly harsh deterrent, or is not otherwise inappropriate for the crime or misbehavior, particularly when less harsh or more appropriate deterrents are reasonably available, it may be employed. Where punishment does not serve as a deterrent, this attempt at justifying it does not apply and cannot be grounds for administering punishment.  It is an empirical question whether punishment in a given instance acts to deter crime or not. Yet it seems that a person who has done something seriously wrong should still be punished, even if it "does no good" for deterring his or anyone else's future bad behavior.

There are two different ways in which punishment may serve as a deterrent when it does.  Some punishments act as a deterrent by making someone fear the consequences of (being caught) committing an act, but there are punishments meant more to impress upon perpetrators the seriousness of their act than to behavioristically scare them into not doing it again.  A loved one's showing profound disappointment or anger may do more to get a person to see that what he did was seriously wrong and ought not to be repeated than it will to make him simply eschew repeating the act out of fear of an unpleasant response.  Similarly with a strong condemnation by a judge prior to sentencing a convicted person, or as his rationale for the sentence being handing down.  One can consider these things to be part of the punishment or as something separate from the punishment.  In some cases, they will be intended for one purpose more than another, the difference being whether the punishment is meant to make the person suffer or to get the person to take seriously his crime or misbehavior and truly understand and realize that what s/he did was wrong.  When punishment is intended to "teach you a lesson," the lesson may be either or both that the behavior was wrong (and thus is something no person who truly wants to be good will do) or that it will not be tolerated without unpleasant consequences for the perpetrator whether s/he cares about what is right or wrong, good or bad, or not.

2) Punishment as being earned and deserved as a fitting consequence to doing evil. In this case, punishment has the same sort of justification as a reward or as thanks and appreciation. It is something that is deserved, based on the actions of the person who has done the deed. It is not "forward-looking" as are utilitarian considerations, trying to find out what will do the most good in the future, but it is "backward looking" in the sense of trying to find out who has done good, or harm, in the past, and acting in some way to fittingly treat those who have done either. 

Suppose, in a simple case, someone steals money from another. Surely s/he should at least have to pay back the money. But I believe s/he should also have to pay for the aggravation and grief s/he has caused as well. Perhaps s/he should have to pay for the efforts of the police to apprehend him/her, perhaps the cost of any incarceration if incarceration is necessary. 

In a crime where what is taken cannot be repaid - murder, rape, battery, irreplaceable property loss, defamation of character, etc. - it seems that something else must be paid besides money, whether it is by the labor of the guilty party or by some other means to try to make up in some way for what was lost.

If we were to act only on utilitarian -- merely forward-looking-- grounds, we would be compelled to reward or bribe people before they have done good things, and punish people before they have done bad things; but once they have done either good or evil, nothing should be done about it because it is "too late" to affect the future. 

Further, on utilitarian, forward-looking, grounds, good people should receive no incentive or attention because they are likely to do good, but people who are about to do great harm should be either coerced or paid not to do it. Incorrigibly good people can be ignored because they need no external incentives to do the right thing.

3) But the strongest argument, I believe, is an extension of the view that no one should be allowed to profit from harm s/he has done others. The actual principle I have in mind is: We ought, as much as possible, to increase the amount of deserved good and deserved joy and contentment in the world, and we ought, as much as possible, to reduce the amount, not only of bad or evil, but of undeserved happiness. We ought to be trying to increase, or at least keep, the balance of deserved happiness over ill and evil, and over undeserved happiness. To that end, anyone who is imminently about to cause, or who has already caused, suffering and sorrow, may have, or should have, at least that much joy and contentment taken from him/her. This is the justification for capital punishment in cases where egregious and irreparable harm has been done. There is no reason to allow someone to experience joy of any sort who has robbed an innocent person of the ability to have such joy. This is also the reason, however, where sincere remorse and repentance over one's actions may make capital punishment unnecessary if the offender's life is filled from then on with abject sorrow and penitence about their past wrong actions, and if they have atoned, or made up to humanity, for the evil they have done by creating at least a commensurate amount of good that only they could do.

It is easy to see that harm may be inflicted on someone who poses a clear and present danger to an innocent person, if such harm is necessary to remove the threat or prevent the evil about to occur, and if the amount or significance of the harm balances (or is proportionate to) the amount of harm/evil to be done. It would be irrational and ludicrous to hold that such measures to prevent evil are justified but are not justified to punish evil that has already been perpetrated. The threat or imminent perpetration of evil is clearly not as bad as the actual commitment of evil, and so what is justified as a deterrent to evil may also serve as a punishment for it. Surely it cannot be right to take the life of someone who is about to murder an innocent person unless it is also right to take that life after they have murdered an innocent person. You cannot hold that innocent life is important only while it is alive; and you cannot then consistently hold it is impermissible to take an innocent life, but it is acceptable and allowable to have taken one. And you cannot hold it is acceptable to use whatever means are necessary to prevent property or other kinds of crimes and evils if you do not hold that such crimes, once they have actually been committed, do not deserve punishment. If crime and evil are important enough to prevent people from doing, they should be important enough to punish people for having done.  But again, whether in preventing evil or punishing it, the amount of force/harm used is only right insofar as it balances as much as possible the harm/evil that is being threatened or that was done.