Some percent of asians, the number varies depending on who's doing the study, but somewhere between one third to one half, have trouble breaking down alcohol. There's a couple of enzymes that are key to breaking down alcohol, and one of those enzymes is defective in some way for those asians. So they flush noticably, which can be embarrassing, might make them throw up with only a little bit, which is also embarrassing, might get a headache/get dizzy, and the effects in general (ie: drunkenness) are stronger and/or last longer.
Likely as a result of this (though it's tough to prove the reason) there's much less alcohol abuse/binge drinking and alcoholism in asian populations compared to other races.
But, and this is pure speculation on my end, it's also probably the explanation for the drunken master. If I like a beer at the end of the day, or some sake, but even one drink will make me flush and a bit dizzy, to an outsider it'll probably look like I'm a drunkard. So people see that, assume asians can't handle their liquor (or drink a ton) and a stereotype is born.