From a utilitarian point of view, the argument against the death penalty would be that it does not increase the overall happiness of society.
Of course, the death penalty brings a great deal of unhappiness to the average person who is subjected to it and to his or her family and friends. It may also bring unhappiness to those who feel that the death penalty is an immoral thing. In order to offset this unhappiness, the death penalty would have to increase the happiness of some segment of the population.
If the death penalty does not deter future murderers, it is not likely to raise the happiness of society as a whole enough to overcome the unhappiness it brings. It will make the family and friends of the victim a bit happier, but it will not bring their loved one back.
The utilitarian argument for the death penalty rests on it deterring potential criminals from acting. If it does not deter future criminals, it does not increase happiness enough to offset the intense unhappiness it brings to a few people.
I feel the argument against a utilitarian justification for executing criminals as described earlier misses out the fact that execution of criminals does bring happiness to a large segment of people though many may not realize it.
Millions of dollars of hard earned money from tax-payers is spent to take care of people while they are in prison. There are many crimes that require people to be imprisoned for a very long duration of time if not their whole lives. This in itself is not going to assure that they do not commit the same crimes again for which they are punished. Would the same money saved by executing them and using it for other causes like providing health care, education, etc. to the general population not provide happiness to them? Also, if examples are set by executing people that committing those kind of crimes is something that is unacceptable and which will not be tolerated it is surely going to make others in the future think twice. That could reduce the large amount of money that is presently spent for law enforcement in other more constructive purposes.
Also, the fact that executing someone who has murdered a person not bringing back the deceased or executing a person who has committed a series of violent sexual assaults not wiping off the horrific memories of those that were assaulted does not mean that the culprits should not be punished. I'm confused by what the previous editor is trying to defend in the response.