Pirates under Roman law, and later law, were considered to be Hostis humani generis, or enemy of all mankind (or perhaps more accurately, enemy of civilized nations). Piracy was essentially a crime that put one beyond all legal protection, and pirates forfeited their natural rights. So to put it bluntly, Romans were not fans of pirates--this is despite the fact that the ordinary conduct of warfare in the Mediterranean is indistinguishable from piracy, but that is a different topic.

So practically speaking, what did the Romans do? The Romans controlled significant stretches of coastline for about a thousand years before 476, so it changed quite a bit over time. In fact, one of our earliest bits of Latin is a treaty between Carthage and Rome in 509 BCE recorded by the historian Polybius. It essentially established zones of control between the two powers and regulations on activities the two could conduct within each others' zones. This treaty would be essentially meaningless unless there was some way each power had a way of regulating their coastline, and the implication behind it is a mutual fear of pirate raids. One could even argue that this treaty was setting out a distinction between legitimate and illegitimate naval activity. So while it is difficult to go into specifics, it is clear that the Romans had some sort of coastal patrol this far back.

There is more story here, but to skip ahead to Augustus we see the earliest development of a dedicated water pacification policy by the Romans. The explosion of the Romans into the eastern Mediterranean during the second century BCE had the effect of shattering the international system of the Hellenistic, with the result that the Greek states could no longer regulate their coasts, and so pirates became extremely rich and powerful, even to the point of raiding Rome's port at Ostia. The beginnings of a solution lay with Pompey the Great's wildly succesful campaign against pirates in 68 BCE. Pompey's success came through an adroit combination of military and diplomacy, and to a point he brought the pirates into the fold--a few decades later, a Cilician pirate was the commander of his son's fleet. This destroyed the ability of "pirate kings" along Turkey's southern coast to wage concerted military action, but pirates remained as parasites on the shipping system. To combat this, Augustus created a dedicated naval command across the Mediterranean to deny the pirates freedom of action. he also reorganized the Roman navy by discarding the clunky, Hellenistic style juggernauts in favor of swifter, sleeker libernian style vessels. This seems to have created a real pacification of the seas, the first and last time such a thing would be achieved until the modern era.

This system held until the Third Century Crisis, a period from 235-285 CE, a period of great turmoil when the empire very nearly broke apart. In particular, the Goths "persuaded" the Greek cities along the Black Sea to let them use their ships, and used them to conduct raids into the very heart of the empire. I'm rather ignorant of this period, however, so I'm not really sure what happened with this. At the end of the period, however, the emperor Diocletian reasserted control and reestablished the naval commands of Augustus.

It is in the period after this, and particularly later in the fourth century, that we can really start to talk about more concrete measures. For one, we have a lot more "on the ground" information--I would urge you to check out /u/shlin28's superb post on slave raiding in Late Antiquity. We also have a first hand account--St. Patrick was a Romano-Briton captured by Irish slave raiding. In fact, Britain gives us a very interesting and fiercely debated piece of information in the form of the Saxon Shore, which was a line of large military fortifications along southern and eastern England and northern France. It is endlessly argued about, but the conventional wisdom is that these were intended to stem increasingly damaging raids from Germany and, possibly, Scandinavia. So, broadly speaking in the later fourth century you can see a roughly simultaneous deterioration of Roman control of the coastlines, leading to the slave raids recorded in North Africa and in Britain. In the North Sea the response was to create a line of fortifications, in the Mediterranean the response is a bit murkier, and seemingly not entirely effective.

So we can see that throughout its history Rome had to deal with the threats that might attack its coasts from the sea, and it gained its most success in doing so by combining diplomacy with an active naval presence in regions of concern. Which is pretty much the case today.

A great and very readable source for ancient piracy is Philip DeSouza's Piracy in the Greco-Roman World.