My answer deals with the navy during the Principate.

First off, it is probably a misnomer to call the seaborne units of the Principate as a "navy" in a modern sense. While a number of "fleets" or classis were stationed across the empire, these fleets were did not form a single, distinctive "navy" organized independently from the "army". All military units within a province was organized into the exercitus of the said province, and transfers between the "Navy" and the "Army" were relatively common place. Legally speaking, all soldiers of the classis were soldier/miles (the same category as that of legionaries and auxiliaries), and not a sailor. The grave epitaphs of fleet soldiers also shows that they referred to themselves as such and not nauta(sailor). Considering these points, it is probably better to see the classis as a specialized seaborne units organized into the overall exercitus of the province, rather than the equivalent of navies of modern states.

With that out of the way, now the details.

There were about a dozen classes during the Principate. The first fleets organized under Augustus were classes Misenensis and Ravennae, each having its HQ in Misenum and Ravenna respectively. They were the largest fleet as well, with the Misenian fleet probably having about 10,000 troops and several hundred warships, and the Ravennate fleet being about half of that size. The "provincial fleets" were gradually formed under the Julio-Claudian dynasty, probably starting from detachments of the two Italian fleets mentioned above or local ships operated by the frontiers legions. Some of the provincial fleets were short lived, but the classes Britannica, Germanica, Pannonica, Moesica, Pontica, Syriaca and Alexandrina are more or less attested continuously from the late 1st century CE to the 3rd century CE. Their size is difficult to guess, but considering that the paygrade of the commanding prefects of these units were equivalent to that of a prefect of a large auxiliary cavalry unit, it was probably not that big, perhaps around 2,000 men and several dozen ships. Their ships were probably smaller as well: the known largest ship of all provincial fleets are Triremes, while the largest known ship for the two Italian fleets are Hexareme.

One problem about the provincial fleets are the lack of sources. AFAIK, not a single mention of them can be found in literary sources, even though many of them existed for more than 200 years. The only source we have of them are epigraphic or archaeological, or in the case the Alexandrian fleet, papyri. This makes it very difficult to analyze what exactly their roles were. The very fact they lasted so long suggests that they had some kind of purpose, but it is difficult to guess what that is. The sources for the two Italian "home fleets" are better, with mentions of them being made in Tacitus and whatnot, and several hundred inscriptions related them to them are found. Yet, we still only have a poor idea as to their daily missions.

A number of epigraphic sources mentions the missions of the provincial fleets, and they don't seem to have anything to do with the sea. One inscription from Lambaesis, Numidia mentions a tunneling work through a mountain in order to construct a aqueduct, and auxiliary troops and a "fleet cohort" is mentioned as workers. Another inscription found in Montana, Moesia Inferior mentions a beast hunt, where a vexillation composed of legionaries, auxiliaries and Moesian fleet soldiers captured bears and bisons for the Imperial games in Rome. Either of them seem unrelated to "proper" naval activities.

So what was the point of these fleets? After all, the Mediterranean sea was Mare Nostrum. Why have fleets if there is no enemy to fight? The problem becomes even more complicated by the fact that the legions operated their own ships as well. If the legions had ships, what was the point of these fleets? This is something scholars have been arguing for decades, and no single answer can be given.

C. G. Starr believed that the fleets were originally founded by Augustus as a deterrent against would-be usurpers, and gradually lost its original purpose by the Pax Romana, thus relegated to menial tasks such as constructions works in order to make them useful.

On the other hand, Michel Reddé emphasized the deterrence role, and that piracy was a problem that could unpredictably occur and the fleets were useful in putting it down.

More recently, Lloyd Hopkins focused on the transfers between fleets and legions and claimed the fleets usefulness as a "manpower pool" for legions. Jorit Wintjes sees the provincial fleets as a specialized seafaring unit, tasked with operating long-distance voyages in the open seas which could not be done by legionary boats.

Either way, one thing is certain: the fleets were there, and for whatever reason the Roman authorities thought it worth enough to keep funding it for 200 years.

Sources

: Octavian Bounegru & Mihail Zahariade, Les Forces Navales du Bas Danube et de la Mer Noire aux Ier-VIe siècles (Oxbow Books, 1996)

: Lloyd Hopkins, "Fleets and Manpower on Land and Sea: the Italian Classes and the Roman Empire 31BC-AD193," PhD. Diss. (Oxford University Research Archive, 2014)

: Michel Reddé, Mare nostrum. Les infrastructures, le dispositif et l'histoire de la marine militaire sous l'Empire romain (École française de Rome, 1986)

: D.B. Saddington, "Classes. The Evolution of the Roman Imperial Fleets", in P. Erdkamp (ed.), A Companion to the Roman Army (Blackwell, 2007), pp. 201-217.

: Chester G. Starr, The Roman Imperial Navy: 31 B. C.-A. D. 324 (W. Heffer & Sons Ltd., 1960; repr., Ares Publishers, 1993)

: Jorit Wintjes, "Sea Power without a Navy? Roman Naval Forces in the Principate," in Marcus O. Jones (ed.), New Interpretations in Naval History: Selected Papers from the Seventeenth McMullen Naval History Symposium Held at the United States Naval Academy 15-16 September 2011 (Naval War College Press, 2015), pp. 13-24.