Disclaimer: I know I'm never going to be President of the United States, nor would I truly want to be. I also know much of what I'm going to say in this hypothetical musing is impractical and improbable to say the least. Get over it.
The POTUS has a lot of power in our system - much more than in most democracies or constitutional republics. Many of the President's powers require congressional rubber stamping, but plenty of them don't. One heavily under-utilized example is the presidential pardon. The President may without rhyme or reason pardon or commute the sentence of anyone convicted of any crime anywhere in the country. Heck, they don't have to be convicted even. You can pre-emptively pardon.
Here is what I would do. Starting on day one of my presidency I would have lists drawn up of every convicted criminal in the country. I would have them categorized into types of offenses and their severity. I would then begin mass pardoning from the least dangerous on upward, with the ultimate goal being emptying our prisons of all but the most dangerous and violent types of criminals.
This would commence in waves for several reasons.
Practically speaking it would be too much work to accomplish in one big push.
A whole bunch of ramifications would result from the sudden influx of unemployed, homeless, former convicts. I would want to give communities time to adapt. Alongside the pardons I would call on aid organizations, churches, and others to get together and strategize how to integrate these people back into society.
I would use the waves as pressure on Congress to demand several things. I would demand legislation aimed at creating jobs and assisting freed prisoners in establishing new lives. I would ask to reorient law enforcement offices around the country in the direction of rehabilitation & prevention rather than punishment. I would demand an end to the war on drugs and changes to federal and state standards for imprisonment - promising to continue pardoning until satisfactory changes were implemented. Each delay would see a new wave of pardons.
I would assemble teams to analyze batches of cases where it would be appropriate to commute sentences rather than pardon if a better result could be achieved. Can murderers and sex offenders be treated psychologically (even permanently in a mental health facility if necessary) rather than imprisoned? Would some benefit from community service? I would commute every single death sentence to a lesser penalty immediately.
Against the threat of impeachment (since this would outrage many people) I would reserve the "nuclear pardon" option to pardon every single prisoner at once on my last day in office if impeached. So long as it remained possible I would continue the slower, more judicious method of pardoning in waves.
By the end of my term I would hope to have very nearly emptied our prisons entirely. Only a small fraction of those currently imprisoned would remain because a better solution could not be found. As such I would hope to crush the prison economy irrevocably. I would relentlessly pressure states and congress to change their laws to utilize alternate forms of punishment beside imprisonment and to create new jobs in social services aimed at overseeing the new procedures of crime and rehabilitation. A few federal prisons would remain as relics for desperate cases where society utterly failed to come up with a better solution.