I can't really see any of my bearers indulging too often. From an economic standpoint, they'd rather not waste their money on something that's relatively frivolous (and, for that same reason, opted not to get tattoos--that, and it'd make them easier for the Seekers to find them). Related to that is the almost-constant need for them to be aware of their surroundings--anything that could impair their judgement or ability to escape is a potential death sentence. Aunt Alice didn't raise a fool for a Bearer.
Now, once they're in a more stable environment, they've probably felt more willing to try some. Evander, for example, is more of a beer-drinker. He's okay with wine, but it's not his favorite (sangria, however, is delicious). I wouldn't be shocked if Athena let him try some wine when he was younger, but only a sip or two every now and them. His first "real" experience was probably with a half-drank bottle of Bud Light at a bus stop one night in L.A. It wasn't the greatest thing he'd ever tasted, and turned him off of drinking beer as something to enjoy for... maybe a couple of months at least, but it managed to stave off the hunger. But once he's in Chicago, things are easier. (As a bit of a self-insert, while he's okay with drinking the occasional Bud or Miller, he was a little spoiled with the craft beer there.)
As for if/when he gets drunk, he's typically more subdued. Not bubbly-drunk or angry-drunk, but more of a drunk that sort of just sits there and relaxes. He'd probably feel sleepy also. Now, if he's already pretty happy, it might make him feel a little more talkative and expressive (he's typically stoic and introverted, although Saint is managing to bring out the more choleric side of him every now and then). If, however, he's in a bad mood... he'll try to see himself out before he explodes on someone--as the son of Poseidon, while he's typically mellow, he'd probably have a similar temper if worse came to worst. Seeing as he already tries to be a decent person, plus the fact that he's the Bearer with a load of expectations (and the sort of luck he's managed to have so far), I highly doubt he'll be drinking too much.
The drinking culture of Ancient Greece was never my specialty when reading or studying history, but I'm pretty sure it was common for those 13 and older to be drinking wine (however watered down it may have been). Given the influence of modern culture, however, there might be a few complications. Children born and raised in America, for example, might have mortal parents that would feel different about drinking at that age. Though, by what little I've read, drunkenness would probably be something discouraged in Olympian society.