It's actually not part of the Eightfold Path! But I will try to answer it from what I understand about Buddhist ethics.
Personally, I don't believe that Buddhist ethics would allow the death penalty. However, there is the death penalty in Thailand, so sometimes Buddhists don't follow their Buddhist ethics.
Buddhism does believe that one is accountable for one's actions. So when one does an unskillful action, then there is usually immediate result, in that, if it is against the law, then it is against the law. It is important to hold people accountable for their actions. Recently, in the Catholic Church there has been a lot of uncovering of sexual abuse and a lot of the church's justifications for allowing the priests to stay priests were that they have to learn how to forgive these priests. But they weren't actually held accountable for their actions and this caused more people to suffer.
According to the Buddhist's ethical rules, if a Monk is discovered to have done such actions, they are immediately disrobed and held accountable. Because the religious order understands that to keep the faith of the people, they must have a certain level of ethical behavior. Monks carry a big responsibility, any religious person carries a big responsibility. A lot of people put trust in that person because they have taken vows to keep to certain ethical behavior. So it's the religious orders' responsibility to make sure that people who go against their ethical behavior are held accountable for that. This they consider would be right punishment.
In Buddhism, for certain actions a Monk is disrobed, for other actions they put on probation and they have to confess their actions in front of the Sangha, which is very humbling for them. So depending on the severity of what they do, they are disrobed. Sometimes they will then be put in jail. There was a recent case where a Monk was found to have committed sexual abuse and not only was disrobed but put in jail. This is important to tell people, that these actions are not skillful and that they won't be tolerated, so people will continue to place trust in the Sangha.
Now, in regular society, it is the law enforcement's responsibility to protect the people, that is compassion for society. People who cause suffering for others have to be held accountable and understand that those actions will bring unbeneficial results for themselves.
There are limits to what is acceptable, certainly torture and physical abuse would not be considered to be right punishment. The death penalty also would not be considered right punishment because then we are taking a life into our hands and doing an unacceptable action to show other people that what they did was unacceptable. Personally, that doesn't make sense to me.
Harming others physically or putting them to death doesn't seem to be right punishment. Certainly, if we understand the law of Kamma, we know that they cannot escape the law of Kamma and it may not be up to us to put them to death. People also have the capacity to change. Understanding the human potential, we understand that a lot of people who had done harmful things in the past had the opportunity to see the deeper, harmful effect of their actions, and they did change.
The most vivid example of this in the scriptures. This Buddhist story is about Angulimala. He was called the "finger bandit". He was a murderer and killed many people. He wore a necklace made out of human fingers which he cut off his dead victims. It had 999 fingers and he wanted one more to make 1000.
His mother realized who the murderer was and was going to try to stop him killing more people. The Buddha through his psychic powers realized that he was going to kill his mother, which was even more serious Kamma for him. So he decided to confront Angulimala himself. He went walking on a path near the forest where Angulimala stayed. Angulimala saw the Buddha and thought to himself, "This is even better, a Monk, I will kill him instead." He ran after the Buddha but the Buddha, through his psychic powers, just kept walking slowly, yet he was going so fast that Angulimala couldn't catch up with him.
This was amazing for Angulimala, he had never been outrun by anybody in the past and he couldn't understand it, since the Buddha appeared to be walking so slowly, but was going so fast. So he called out, "Stop, stop, Monk!". The Buddha said, "I have stopped, Angulimala, it is you who has not stopped." Angulimala couldn't understand, "What do you mean, stop?" The Buddha said, "I have stopped killing living beings, but it's you who hasn't stopped!" At that moment, Angulimala understood his wrong lifestyle. He became so overwhelmed and awed by the Buddha that he became a follower and wanted to be ordained as a Monk. The Buddha ordained him.
Yet, of course, developing his mind in meditation practice was extremely difficult for him because every time he tried to meditate he had these visions of all these people he had killed. He was having a terrible time. On alms-round all the villagers would see him and realizing who he was, they would throw sticks and stones at him. They wouldn't kill him because he was a follower of the Buddha now, but they kept beating on him whenever he came around. He would come back bleeding. The Buddha said, "Have patience, it's your Kamma fruiting." He continued having a terrible time meditating. Finally one day he was going into town and heard a woman in labor crying. Seemingly for the first time in his life he felt a sense of compassion arise and he wanted to help this woman.
He went back to the Buddha and told the Buddha about her. The Buddha said, "Go and tell the woman, that in your life you have not harmed a living being. And through the power of that truth, may she will deliver her baby safely." Angulimala replied, "How can I do that, when I have harmed so many people in my life?" The Buddha explained that what he meant, was since Angulimala has been ordained he has not harmed a living being.
He went and told the woman this statement, and she had a safe childbirth afterward. He was then a different person, he understood he could change. He realized he had to work very hard in that life to become enlightened otherwise he was going to an unhappy destination. So he worked very hard and he became fully enlightened.
This is a very inspiring story concerning people who have harmed others and their possibility for change.