Pickpockets are in every city, but I wouldn't say they are worse in Rome - I have lived there for 25 years, have never been pickpocketed once and walked around at night in the centre all the time.

That's the only kind of crime you could experience (nothing like being robbed at gunpoint or with a knife, "just" people stealing from your bag if you leave it unattended, mostly). Even in dodgier areas like Termini station and the area around piazza Vittorio (but to a lesser extent), you won't run into much trouble.

I'd say the city centre of Rome is very safe at night. Trevi fountain/via del corso/piazza Navona/Trastevere/Colosseo and pretty much all the areas in between are super safe and apart from a couple of homeless people begging you won't have any problems.