Actually, the future that some fans describe, with massive unemployment and aimlessness and people looking for enlightenment, I could easily see incredibly high crime rates and social problems and drug use.
Billions of people with really nothing to do except get into trouble.
You could decriminalize most offenses, or ship vast numbers off to penal colonies/penal planets (therefor "no crime").
This is why I made sure to add accountability in there. Irresponsible people in Trek (at least, irresponsibility stemming from sloth) are a vast minority; even civilians who have little to do with Starfleet operations end up being pretty ambitious, like Joseph Sisko's restaurant, or freighter captains, or various civilian scientists. That's not to say that a world with minimal prisons or no police means a system of rampant misbehavior and no accountability at all; we clearly see wrongdoers getting punished in Trek, it's just not as widespread as today because sense of accountability is developed, through community or concern for the next person or other means. Their social order is built on fulfillment of personal goals. Order isn't maintained through brute force and threats of violence, but through the collective sense of conscience and empathy (taught through parenting, education, friends, family, neighbors, etc), with law enforcement (whatever that may truly look like in this era) complimenting instead of substituting that sense. The systemic root problems of crime would be removed.
You wrong someone or you break the law, you get punished, obviously. We've seen that in Trek many times over. Nothing wrong with that. What we haven't seen is mass incarceration and incarceration for the sake of profit rather than order, both of which is extremely damaging today both to the fabric of society and to the drain of resources (which, in turn, perpetuates class and race divides, rather than promoting unity, stability, and rehabilitation). On the contrary, we've seen people in Trek serve their time, but the big difference here is that prison time helps rehabilitation, perspective, and even remorse and actively working to make up for their misdeeds to society, whereas modern-day incarceration means stigmatization and heavy debt, and thus huge difficulty in getting honest work or social support, which often leads to repeat crime or destabilization of communities (and thus higher crime rates). The Netherlands and Sweden are good examples of showing how a different approach to prison thanks to plentiful resources can happen. In the US, Naperville, IL -- which invests much more in its social programs than neighboring suburbs -- is not only wealthier (thus more resources), but also has a lower crime rate (even though criminals could easily come from those suburbs and Chicago), and is seen as a positive model of police reform as an adaptation to the social factors outside the police itself.
In my head canon, I believe whatever system of law enforcement and justice they have, is more invested in restorative justice rather than strictly punitive like most of today, because the motivations between both systems would be vastly different based on the needs of the future.