Actually, a police force for the city of Rome was created by emperor Augustus, which would still fit into your time period: the Vigiles Urbani.

They also acted as firefighters, and were organized somewhat like a Roman legion, with seven cohorts of seven centuries each.

Burglary and robbery must have been a problem for the Romans, for this were the crimes the vigiles were supposed to look out for at night.

We also know that piracy was a substantial problem: Julius Caesar himself was kidnapped by pirates as a young man, and held for ransom. Kidnapping victims whose families who could not pay ransom were often sold into slavery.

Pompey owned much of his reputation as a hero to a year-long, successful anti-piracy campaign he conducted with the Roman fleet in 67 BC.