I lived in NYC from 1986 - 1995, and had a very different experience to the ones described here.
During the beginning of my stint there, crime was rife. Times Square and the surrounding area was scary, and the 'borderline' neighbourhoods were truly 'no go' areas. Gang activity was common, and crack cocaine was beginning to be a big problem for many on the fringes of society - it was plentiful, cheap, and many people became addicted and so muggings (to fuel the habit) became common.
The mayor and police began a 'zero tolerance' policy. The thinking was that enforcement of the 'petty offense' laws would lead to a huge reduction in crime overall. There was robust enforcement for fare-dodging, for breaking windows, for loitering, for drug possession, etc - basically any 'minor' crime was punished harshly.
The officials believed that petty crimes often led to bigger crimes and that the petty criminals were often the same people committing the more serious crimes. By arresting and incarcerating people for petty crime, the more serious crime rate dropped significantly. And, it seemed that the criminals quickly began to think the punishments were so swift and harsh that it simply wasn't 'worth it' to commit low-grade offenses.
What I can't tell you is what became of that generation of young offenders - did they eventually decide to 'go straight'? Are they in jail for different offenses? Who knows.
What is true is that NYC is a much safer place since the policy was implemented.