There are no teachings in Buddhism that promote the death penalty, but some Buddhist countries do still use the death penalty. Reasons for this are:
The death penalty is clearly against Buddhist teaching. Buddhists place great emphasis on non-violence and compassion for all life. The first Precept requires individuals to abstain from injuring or killing any living creature.
The Buddha did not explicitly speak about the death penalty but his teachings show no sympathy for physical punishment, no matter how bad the crime.
An action, even if it brings benefit to oneself, cannot be considered a good action if it causes physical and mental pain to another being.The Buddha