Both. By raising the cost (punishment * chance of getting caught), you prevent it. Sure, many of those who flaunt the law have to learn the hard way, but there's no question raising either or both of those inputs will reduce the overall amount of crime.
I don't believe the death penalty does much to deter murder or is cost-effective (although I still consider it just in most circumstances), because the perpetrator often doesn't value their own life that much. When you have to live with the punishment, I believe it is a deterrent for most, analogous to the prospect of living with a lot of debt: the price of stuff affects our behavior.
That's irrelevant; clearly some people change their behavior due to the risk of breaking the law. However, unenforced rules (ie driving 56mph in a 55mph zone) are generally ignored.
I don't believe the death penalty does much to deter murder or is cost-effective (although I still consider it just in most circumstances), because the perpetrator often doesn't value their own life that much. When you have to live with the punishment, I believe it is a deterrent for most, analogous to the prospect of living with a lot of debt: the price of stuff affects our behavior.