Athenian (and other Greek) galleys (a trireme, or later quadrireme and quinquereme) were commanded by a Trierarch. Generally, the trierarch was chosen, or volunteered, because he was wealthy enough to pay for the cost of fitting it out a ship and maintaining it for the current year. This was the civic duty of a wealthy Athenian, and a matter of pride, although sometimes grudgingly performed. If the trierarch wasn't able or willing to command ship, he could delegate that authority to his kybernetes, or second in command.

Lionel Casson, Ships and Seamanship in the Ancient World

At Athens (the only Greek state for which we have good information on this sort of thing) a ship was commanded by a trierarch (τριήραρχος, from τριήρης, trireme+ἀρχός, leader). Trierarchs were private citizens of particular wealth who, in typical Athenian fashion, were required to use their private funds to maintain a ship for the state. I say typical Athenian fashion because the Athenian state didn't really have taxes, except for emergency taxes on the very wealthiest (the εἰσφορά) and a relatively small tax on goods going through Piraeus (as well as a couple other cities during the Peloponnesian War). Most funding came in the form of tribute from the Athenian Empire, but a great deal of public works were funded through the liturgy (λειτουργία), the service of the wealthy on behalf of the state. Wealthy citizens subject to the liturgy were required to provide particular duties for the state out of their own pockets. These services included things like the raising and training of a chorus during the dramatic performances, providing banquets for one's tribe during festivals, etc. Military duties included the maintenance of a horse for war and, from 483 on, the maintenance of a trireme. At the very latest, if not before, the old system of using tribal leaders to command the fleet (the ναυκραρία) was abolished in 483, though probably it was done away with during the reorganization of the tribes after the expulsion of the tyrants.

Trierarchs were required to equip and maintain their ships for the duration of a year. The state provided the ship and the crew, which was usually paid with state funds, but the trierarch was responsible for the costs of maintenance and had to command the crew. Trierarchy was expensive and trierarchs were often called up for several years in a row--in 411 a reform was introduced that provided for two trierarchs per ship to split the cost. In 357 the system was further reformed. Under this new law the symmoriae (συμμορίαι), taxation groups of the wealthiest citizens (100 groups total) which had been established in 378, were extended. The new law grouped the 1200 wealthiest citizens into 20, not 100, symmoriae and the cost of the trierarchy was divided evenly among the members of each symmoria. In 340 Demosthenes further reformed the trierarchy by scaling the cost of the trierarchy according to the assets of the individuals within each symmoria, so that while all 1200 were providing for the trierarchy through the liturgy the majority of the costs were falling on the very wealthiest two or three hundred. As it became harder and harder to find people with enough wealth for the liturgy in the 4th Century the liturgy in general came under fire, and under Demetrius of Phalerum the liturgy was abolished entirely.

Roman naval commands worked completely differently, however. As with armies fleet commands were generally given to magistrates and promagistrates above praetorian rank, although sometimes we see extraordinary magistracies being created for the command of a fleet, often with other duties attached to it. However, like in the army, the use of prefects in the fleet was common beginning in the Republic. In the Principate the praefectus classis was established as a particular type of prefect. There were two praefecti classis, one commanding the praetorian fleet at Ravenna and the other commanding the (senior) praetorian fleet at Misenum, as well as a number of provincial prefects in command of the various provincial fleets.

But those are fleet commanders, and I only mention them because the way that the Romans dealt with military magistracies is totally different from the Athenians, who basically had their στρατηγοί and nothing else to command both armies and fleets. The Roman fleet's organization mirrored that of the army. All marines and crewmen of a Roman warship were considered miles, soldiers, although they were junior to their landlubber counterparts. Just like the army the entire crew of a warship, regardless of its size, was organized into a century and commanded by a centurion. A warship of decent size apparently would also have an optio on board, as well as a number of lesser beneficiarii and immunes--it's from these that I have to presume most of the crew other than the rowers came, but I'm not sure if we actually know that.