Probably nothing, at least in terms of prevention.

As a preliminary, it should be noted that Roman society as a whole was likely not notably more promiscuous during the Late Republic than in any other period. Some members of the Roman aristocracy - people like Clodius and his sister Clodia, or even that bald adulterer Caesar - did indeed flout the conventions of traditional morality; and to judge from the bohemian poems of Catullus (or the sour moral commentary of the historian Sallust), such behavior really was seen as unprecedented and shocking. But we shouldn't imagine anything like a Roman "sexual revolution" taking place in these years.

That disclaimer declaimed, on to the actual question. The Romans certainly suffered from a few familiar STDs, though the vagueness and generality of the symptoms mentioned by medical authors make clear diagnosis difficult. It seems tolerably clear that herpes and chlamydia were present; the jury (as far as I know) is still out on gonorrhea. Yet while the Romans had at least a vague sense that diseases could be sexually transmitted (as several medical texts attest), there is no evidence that they used any kind of prophylactic against infection. A wide array of more or less ineffective contraceptive methods were practiced, but these were all the woman's prerogative, and followed intercourse.

We know a little more about the treatment of STDs. The Roman encyclopedist Celsus, for example, provides a long list of herbal remedies and poultices that can be used to treat genital sores - e.g. "If a free discharge of foul pus begins, the ulcers should be bathed with lentil gruel to which a little honey has been added..." (6.18.2). Celsus goes on to provide some rather harrowing advice about the amputation and cauterization of infected tissue.

I wish I had more to share. But thanks to the Romans' ignorance of germ theory and the nature of our literary sources, there isn't much more to say about the topic.