Yes, they did, for the purpose of controlling pirates. I wrote a fairly longish post on Roman anti-piracy measures a while back that I am rather fond of, and it goes into the details of the various different ways that Rome tried to control (or not control) pirates--also a bit about what "pirate" means in an ancient context.

It is worth noting that Rome's navy was structured around very different needs post-Actium--given that there was no credible military threat, they had no need to build the large Hellenistic style mega-shps, instead favoring the sleek well made Liburnian style. You could somewhat simplistically sum it up as the transition from a navy to a coast guard.

Rome essentially let her navy decay after the Punic Wars, often recruiting or conscripting ships from allies in emergencies until the Civil Wars, when the contestants again built up large navies. Octavian (Augustus) organized the first permanent Mediterranean fleet from a portion of the ships that had been used in the Civil Wars, and the Empire maintained a permanent fleet until it vanished in the third and fourth centuries (although the Eastern Empire continued to maintain a navy). The navy was used to keep the Mediterranean clear of pirates.

Sauce: OCD second edition

Presumably the navy also helped move troops and quell rebellions, although the OCD didn't really mention that part.