Following the defeat of the Seleucid Empire, the Roman navy was essentially wound down. The navy had been, for the most part, first built up in numbers and maintained to counter the threats of the Carthaginians, and the fleet that took part in operations against first the Macedonians then the Seleucids was this force. We have to remember that, at this time, the Roman armies were still 'citizen' forces, and this included the men who manned the fleets. Manning and maintaining them was a great cost.

Now, there is no direct source, as far as I know, of the exact fate of the ships of the Roman fleet, however, since warships were not best suited to trade purposes, they are unlikely to have been sold. We can expect that they would have been laid up, and possibly broken up for scrap materials such as wood. Without moneys being applied to maintain them though, these ships would quickly become unseaworthy.

Now, this merely talk about the ships used by Rome. Allies such as Rhodes would have retained their fleets, and indeed, Rhodes became responsible for protecting trade in the Eastern Mediterranean as they were the primary benefactors in the early 2nd Century BCE of Roman dominance. If any state purchased or was gifted Roman military vessels, it would have been Rhodes.

The one caveat to this was that, less than a generation later, Rhodes so feared Roman power that they went to the length of executing citizens Romans associated with a betrayal (in the Third Macedonian War) rather than risk conflict with Rome. As anyone familiar with history, particularly medieval history, knows, Rhodes is not an easy place to invade. It's impossible to say if thiss fear was because Rhodes feared existing Roman naval capabililities (thus suggessting the existence of at least a reserve naval capability capable of being quickly manned) or the understanding that Rome could and would build new ships to quickly overwhelm Rhodian naval forces is hard to say. However, evidence points likely more to the later, as in the First Century against Mithradates, Sulla had to scrounge up ships from wherever he could get them to face off against a naval force likely far smaller than what the Rhodians at their 2nd century height could muster to oppose an invasion, suggesting both the absence of a naval reserve, as well a the demonstrated ability of Rome to find a way (something they demonstrated repeatedly in their history).

Given that last bit of information, it's very unlikely those ships would have been 'saved' or investment made into retaining them, given the political situation in the Mediterranean at the time. Rome had never been a naval focused power, and with the threats from everywhere in the Mediterranean seemingly crushed by 189BCE, the most likely fate of those ships was to be sold or destroyed.