An excellent article from Crime & Consequences Blog about the proof.

An excerpt.

No one doubts that the increased use of mandatory minimum sentencing laws has contributed, likely significantly, to the big increase in the prison population from 1990-2010 (it has dropped since then).

The next question crucial to our sentencing debate, then, is how much has increased incarceration contributed to the astonishing drop in crime over those 20 years (and astonishing is the right word, see this table showing that crime rates fell by nearly half).

Obviously, if increased incarceration accounts for only a small part of the falloff in crime, then the case for easing off on the use of prison becomes stronger. Conversely, if more prison has been a big driver of plummeting crime, the country justifiably will be more hesitant to go back to the softer policies that brought us the crime explosion in the quarter century before 1990.

This central question has been studied.  How much does increased incarceration contribute to the drop in crime?  What do the data say?

It tuns out we have plenty of data and they say pretty much the same thing.

The most systematic study is here, in which Prof. Steve Levitt of Chicago finds, in a 2004 research paper, that increased incarceration accounted for 25% or perhaps more of the large decrease in crime in the Nineties, and noting that the “evidence linking increased punishment to lower crime rates is very strong.”

Prof. William Spelman at Texas found the amount to be between 27% and 34%, see here.

Eminent UCLA criminologist Prof. James Q. Wilson took a look at the Levitt and Spelman conclusions and agreed with them in a 2011 article in the Wall Street Journal.

Finally, a well-regarded expert favorable to  sentencing reform, John Malcolm of the Heritage Foundation, acknowledged in his 2014 debate with Judge Michael Mukasey and me that increased incarceration may account for between 25% and 35% of the reduction in crime in the relevant period. See John’s balanced presentation between 7:30 – 8:10 of the tape of his remarks.

So the consensus of neutral scholarship is that increased use of incarceration resulted in roughly 28% of the crime decrease that started at the dawn of the Nineties.and went on for the next 20 years (or more).