Excluding the Hogwarts teachers already covered in the other answer, we do see a few other characters in their cups;
Hagrid & Prof. McGonagall (PS)
Flaming Christmas puddings followed the turkey. Percy nearly broke his
teeth on a silver Sickle embedded in his slice. Harry watched Hagrid
getting redder and redder in the face as he called for more wine,
finally kissing Professor McGonagall on the cheek, who, to Harry’s
amazement, giggled and blushed, her top hat lop-sided.
Aunt Marge (and probably Mr and Mrs Dursley) (POA)
At last, at long last, the final evening of Marge’s stay arrived. Aunt Petunia cooked a fancy dinner and Uncle Vernon uncorked several bottles of wine. They got all the way through the soup and the salmon without a single mention of Harry’s faults; during the lemon meringue pie, Uncle Vernon bored them all with a long talk about Grunnings, his drill-making company; then Aunt Petunia made coffee and Uncle Vernon brought out a bottle of brandy.
‘Can I tempt you, Marge?’
Aunt Marge had already had rather a lot of wine. Her huge face was very red.
‘Just a small one, then,’ she chuckled. ‘A bit more than that … and a bit more … that’s the boy.’
Sir Cadogan (PoA)
When they reached the portrait hole they found Sir Cadogan enjoying a Christmas party with a couple of monks, several previous Headmasters of Hogwarts and his fat pony. He pushed up his visor and toasted them with a flagon of mead.
‘Merry – hic – Christmas! Password?’
Irish Quidditch fans (GoF)
By the light of the few fires that were still burning, he could see
people running away into the woods, fleeing something that was moving
across the field towards them, something that was emitting odd flashes
of light, and noises like gunfire. Loud jeering, roars of laughter and
drunken yells were drifting towards them; then came a burst of strong
green light, which illuminated the scene.
He thought the Beauxbatons caravan was likely to be pretty chilly, too. Hagrid, he noticed, was keeping Madame Maxime’s horses well provided with their preferred drink of single-malt whisky; the fumes wafting from the trough in the corner of their paddock were enough to make the entire Care of Magical Creatures class light headed.
Morfin Gaunt (HBP)
‘YOU!’ he bellowed. ‘YOU!’
And he hurtled drunkenly at Riddle, wand and knife held aloft.
Riddle spoke in Parseltongue.
The Fat Lady & Violet & Monks (HBP)
‘I’ve got something for you, Harry,’ said Hermione, neither looking at Ron nor giving any sign that she had heard him. ‘Oh, hang on – password. Abstinence.’
‘Precisely,’ said the Fat Lady in a feeble voice, and swung forwards to reveal the portrait hole.
‘What’s up with her?’ asked Harry.
‘Overindulged over Christmas, apparently,’ said Hermione, rolling her eyes as she led the way into the packed common room. ‘She and her friend Violet drank their way through all the wine in that picture of drunk monks down by the Charms corridor. Anyway …’
A drunken Weasely uncle (DH)
Wandering through the crowd so as to escape a drunken uncle of Ron’s who seemed unsure whether or not Harry was his son, Harry spotted an old wizard sitting alone at a table.
The Oldest Brother (DH)
‘“That very night, another wizard crept upon the oldest brother as he lay, wine-sodden, upon his bed. The thief took the wand and, for good measure, slit the oldest brother’s throat.
Bill, Lupin, Fleur and Harry (DH)
Bill had soon filled all of their goblets; they stood and raised them high in a toast.
‘To Teddy Remus Lupin,’ said Lupin, ‘a great wizard in the making!’
‘’Oo does ’e look like?’ Fleur enquired.
‘I think he looks like Dora, but she thinks he is like me. Not much hair. It looked black when he was born, but I swear it’s turned ginger in the hour since. Probably be blond by the time I get back. Andromeda says Tonks’s hair started changing colour the day that she was born.’ He drained his goblet. ‘Oh, go on then, just one more,’ he added, beaming, as Bill made to fill it again.
The wind buffeted the little cottage and the fire leapt and crackled, and Bill was soon opening another bottle of wine.
Harry (again) and Ron Weasley (DH)
‘Oh,’ said Ron. ‘Yeah … well, I’m hungry!’ he added defensively, as his stomach gave an enormous rumble.
‘I got food,’ said Aberforth, and he sloped out of the room, reappearing moments later with a large loaf of bread, some cheese and a pewter jug of mead, which he set upon a small table in front of the fire. Ravenous, they ate and drank, and for a while there was silence but for the crackle of the fire, the clink of goblets and the sound of chewing.
‘Right then,’ said Aberforth, when they had eaten their fill, and Harry and Ron sat slumped dozily in their chairs.
Various hooting muggle idiots (DH)
‘Where else is there?’ asked Hermione, cringing as the men on the other side of the road started wolf-whistling at her. ‘We can hardly book rooms at the Leaky Cauldron, can we? And Grimmauld Place is out if Snape can get in there … I suppose we could try my parents’ house, though I think there’s a chance they might check there … oh, I wish they’d shut up!’
‘All right, darling?’ the drunkest of the men on the other pavement was yelling. ‘Fancy a drink? Ditch ginger and come and have a pint!’
Goodwin Kneede (QTtA)
Radulf got a Blooder in the head because old Ugga wasn’t quick enough
with his club. The new scoring barrels worked well. Three at each end
on stilts, Oona from the inn gave us them. She let us have free mead
all night because we won as well. Gunhilda was a bit angry I got back
so late. I had to duck a couple of nasty jinxes but I’ve got my
fingers back now.
Ron Weasley (CC)
RON: Well, we were only young when we did it the first time and I got very drunk and — well, to be honest, I can’t remember much of it
and . . . The truth is — I love you, Hermione Granger, and whatever
time says — I’d like the opportunity to say so in front of lots of
other people. Again. Sober.
Ron and Neville (CC)
RON: I was having a couple of firewhiskies with Neville in Hogsmeade — as you do — setting the world to rights — as we do — and
we were coming back — quite late, very late, and trying to work out
which Floo I could use because when you’ve had a drink sometimes you
don’t want to use the tight ones — or the turny ones or —