1). Some crimes are so serious that only the death penalty will suffice. Violent and sexual crimes against children, for example. Those who commit such heinous acts will always represent a danger to society. Thus the death penalty is the only appropriate punishment in such cases.
2). The death penalty acts as a deterrent. If people know that they are likely to be executed for the most serious of crimes, then they will be less likely to commit them.
3). The death penalty saves taxpayers' dollars. In comparison with all the money spent on prisoners' upkeep, execution involves relatively little expense.
4). With the best will in the world, some criminals are beyond rehabilitation. Whether they're out on the streets or locked up inside a prison they're always going to represent a danger to someone. The death penalty deals with this problem by removing dangerous elements from society permanently.
5). The death penalty satisfies the natural human desire for vengeance. This may not be one of its stated aims, but it's the main reason why people support it. Executing criminals takes vengeance out of the hands of private individuals, thus maintaining social stability while at the same time satisfying the community's desire for revenge.