The problems of the drunken master prestige class are many. Actually, it has almost nothing about it that isn’t just about completely terrible.
Very harsh prerequisites
Qualifying for drunken master is miserable: you need two awful feats, plus one of the taxiest of feats in the game in Improved Unarmed Strike. This is a huge knock against the drunken master.
Worse, it requires flurry of blows and evasion: that makes it hard to be a drunken master without being a monk. That’s bad because monks are bad, but more significantly, that’s bad because drunken master is really bad for monks in particular. While it uses unarmed strike damage for its improvised weapon attacks, it doesn’t progress monk unarmed strike damage so that doesn’t add very much. And then it goes and tanks Wisdom, which is pretty crucial for... nearly every monk class feature of note.
Drunk like a demon is a really bad version of rage
Consider rage: +4 Strength and Constitution, +2 to Will, −2 to AC, which can be activated as a free action. In comparison, a second-level drunken master +4 to Strength or Constitution, while taking −4 to Intelligence and Wisdom (after using it twice). That tanks your Will save, which is often critically important, kills monk class features, and it can drop you out of qualifying for feats. And it takes two turns to build up.
Higher levels allow larger bonuses, but remember that the barbarian started from 1st, while the drunken master started at 6th. Barbarian 8 gets you +6/+6, which the drunken master matches at the same ECL. And the larger bonuses take longer and longer to accrue, which is a really bad thing in a fight. The barbarian’s are still a free action.
Improvised Weapons are a really bad idea
Improvised weapons, by definition, cannot be magically enhanced. This is a huge flaw that the drunken master has no good answer for. Even when they do d12s, you can just use a greatsword, whose 2d6 is superior to 1d12, and magically enhance that. You get to add on your unarmed strike damage but it’s not nearly enough since drunken master doesn’t even progress monk unarmed strike damage. And it’s incompatible with any unarmed-strike-specific features you might have, so even that route is denied to you. Neither amulet of mighty fists nor necklace of natural attacks can be applied to improvised weapons. What all this means is, even as a drunken master, an option you could have used by default is better than your iconic class feature. A drunken master is seriously better off using real weapons than improvised ones.
The rest of the features (barring stagger) are really mediocre
They just don’t do much. Corkscrew rush is almost good, but the risk of proning yourself is just awful. The bonus feats are not exactly high-quality, and you could have gotten them long ago as a fighter or something. The AC bonuses are really small. Breath of flame is barely better than flaming hands, a 1st-level spell.
Stagger is quite good, but almost-as-good effects are available far more cheaply
Stagger allows you to charge in any twisting path you want, and allows you to negate all attacks of opportunity with a single DC 15 Tumble check, which is pretty awesome.
But feats exist that allow you to make one turn during a charge; you could take a couple of those, and it’ll be really rare that you can’t charge who you want to. And Tumble can just be used during a normal charge; you have to make a Tumble check for each attack of opportunity, but Tumble is worth being good at.
Note that this has nothing to do with whether or not playing a drunken master character is a good idea
A drunken master character could be a lot of fun—but you don’t need, or want, the drunken master prestige class to do it. Build any martial artist sort of character you want, and then describe yourself as using the drunken fist style. After all, the actual drunken fist martial art, zui quan, does not make significant use of improvised weapons. (Or even alcohol. They're just imitating the unpredictable movements of a drunkard because it gives them an edge in a fight.) Zui quan is a mostly barehanded fighting style, with some staff or sword using forms. Those are real weapons, not improvised ones. So go ahead and play a drunken master if you like—but skip the actual prestige class, you’re better off without it.