It has it's merits, but I don't think it's workable.
Firstly, I strongly dislike the idea of being an adult at 18 - legally able to vote, buy cigarettes etc. but not to drink.
Secondly,of course it wouldn't stop all teenage drinking. However, it could help to reduce or at least change some of the excess drinking in dangerous situations. A higher drinking age would mean fewer kids getting drunk in venues, looking for fights, out in traffic etc. The underage drinking would be more concentrated in peoples' homes.
Thirdly, the culture. As others have said, our issues with alcohol are mostly around our culture. Partially, the drinking culture - partially, violence/disrespect for others in our culture. If we had a higher drinking age, maybe that would go some way toward de-normalising drinking amongst teenagers. I can't see that it would do enough though. The reality is though, it's NOT going to happen. There'd be so much opposition - both in the form of promoting individual rights, and the alcohol lobby.
Sure, alcohol consumption can have some serious health consequences... but I don't think that's where our focus really needs to be. Soft drinks can have some pretty serious health consequences too.... good luck getting teenagers to avoid them on that basis alone. The more serious consequences of alcohol are related to violence, sexual assault, driving under the influence etc. I think that's where our focus needs to be. On education, awareness, social policies etc.