Punishment is not effective for most people in reducing crime. One thing that is effective is increasing the perceived risk of getting caught. People rarely commit crimes when they are sure they will be caught.
Even in the worst case, recidivism rates only reach around 2/3. That means punishment (incarceration) reforms criminals 1/3 the time. When you consider all of the problems with prisons in the U.S., and all of the hurdles offenders have to pass in order to successfully reintegrate into society, it’s quite remarkable that so many manage to do so.
There are plenty of medical interventions, particularly for mental illnesses, that would dream of 1/3 effectiveness. Now consider the fact that recidivism rates are often much lower than 2/3, and in many cases are closer to 1/3. That tells me that punishment actually is quite effective at reducing crime.
Correlation is not causation. If punishment is causing the recidivism then you'd expect an increase in punishment to increase recidivism. It doesn't. You also need to compare prevention with recidivism. Optimal approaches prevent the crime in the first place.
For the sake of argument, let's assume that punishment has absolutely no effect on whether someone will commit a crime. Even so, there's still a benefit to imprisonment: it separates criminals from society for a period of time, during which they won't be committing crimes against it.
To be clearer, it's not that some form of punishment isn't effective. It's that more severe punishment is not more effective. And prevention is the most effective. There are plenty of studies. A search doesn't surface any for you? What search terms have you tried?
There are plenty of medical interventions, particularly for mental illnesses, that would dream of 1/3 effectiveness. Now consider the fact that recidivism rates are often much lower than 2/3, and in many cases are closer to 1/3. That tells me that punishment actually is quite effective at reducing crime.