Changing the system locale just to get that to display properly isn't the best of solutions though it works. Unless you read Korean often and use Korean programs often there are other solutions!
1) If you have to use WinAmp (I recommend another client read below)...you can try to use AppLocale (and it's updated version pAppLocale). What it does is it allows you to run a program under a different locale without the need to change the system locale. pAppLocale fixes some minor bugs/annoyances. You'll have to do some figuring (system's advice seem good) of WinAmp though and it doesn't always work properly. So this is a temporary fix that doesn't quite fix the whole issue.
2) Switch to another player....iTunes I believe supports UNICODE but iTunes uses up too much memory and has too many unnecessary stuff + doesn't looks as great (at least when I last used it). What's good about iTunes is that it had a great playlist management (at least that's what I often hear). I use foobar2000 which is a fantastic UNICODE-supporting player. It's got tons of features (like gapless playback, freedb auto-tag, masstagger, etc) and has low memory footprint. The problem is I haven't been able to find a private Korean freedb server (I've only found one for Japanese music but it does have some more widely known Korean albums on there as well).
Proof that WIKIs are really really wonderful --->
Comparison of media players <-- read this and you'll find a player suited for you