Binge drinking might well have always been a problem, but it seems to me that there is now more concrete evidence about the links between excessive alcohol consumption and health concerns (both physical and mental health). That's probably why it's come to the general attention. Youngsters are quite happy to flaunt their alcoholic habits, and it's become the issue of the moment to fix. Once upon a time it was smoking, then it was drink driving, now it's binge drinking.
On the issue of how to go about fixing it:
Upping the drinking age will do more harm than good. As soon as you make something illegal, it immediately increases the attraction, especially for rebellious youths. Neither should they make the legal age any younger. Leave it at 18, it's as good an age as any, but change all the other aspects of the youth drinking culture.
Warnings on bottles might dissuade a few, so let's go for it. One approach isn't going to fix it, and it's a huge problem, so we have to do everything. Warning labels are one tiny aspect of what we can do.
Ban those sweet lolly-water Cruisers et al, which are very attractive, especially to the young ladies, and are nearly always more than one standard drink. It's like drinking soft drink; you don't even notice that you're drinking until it's too late.
Crack down on vendors not checking ID. Crack down on venues selling alcohol to people who are already drunk; it's illegal yet it happens all the time, because some are more concerned about making a buck than the kids losing brain cells. Crack down on people who produce fake IDs. As Yak said, the threat of the law works for a lot of kids, and really is the way to go. You'll always get the rebellious ones who don't give a shit, but it will scare a lot of kids. The police are pretty stretched as it is, but anything they can do to make their presence felt is vital. And the Liquour Licencing Commission has powers regarding vendors, so they can help out too.
Widespread education in high schools (and universities). Scare the shit out of them; give them gruesome detail about brain injury, foetal alcohol syndrome, drink driving, anything and everything that might dissuade them.
Parental/family influence is the hard one. Different kids take parental advice differently; for some, promoting moderate drinking at home will do the trick, for some it will only give them a taste for it which will escalate. Some kids might see the prohibition of alcohol at home as an excuse to go out and get smashed because they haven't experienced it. That's the difficult issue, and that's probably just about the crux of the issue. I don't have any answers there!
There's no one way that's going to stop this, it's about gradually developing a culture that doesn't revere binge drinking in quite the same way. It might take a number of years, but it's worked with smoking. People still smoke, and people still take up smoking, but it's hugely improved in the last 20 or so years. There's no reason why the same thing can't happen with alcohol.