You basically have two options: (1) GOMPlayer or (2) manually download them yourself.
1. GOMPlayerGOMPlayer is by far the most popular desktop video playing software among Koreans. I'll be honest: I've never gotten GOMPlayer to work properly when it comes to its feature of automatically downloading Korean subtitles. My friends say they just load the video and something pops-up saying "ooh you want Korean subs for this? YOU GOT IT BUDDY" but I've never had such luck. Maybe they are using the Korean-language version of GOM (I use the English), or a newer version, or a magic version. Whatever. I'm stuck with the manual route. Not like it's that hard anyway
By the way, GOM (곰) is Korean for "bear" thus the paw icon, and although I can't get me some automatically downloaded subs, I still choose it for watching the videos once I've got them, as some apps (I'm looking at you, VLC) seem never to get the encoding right. GOM Player has always displayed the subtitles correctly the very first time.
2. Manual downloadingFor manual downloading, ignore any English-langauge sites; only the Korean sites will have a decent selection. There are many out there, but my go-to site is Cinest. Add the title of the movie you're looking for in the search box (the smaller one, next to 제목 ("title")) and hit search. A list of subtitle files comes up. Click one, and download the .smi or .srt file on the resulting page. I usually download 2~3 as the timing will be off on some. Load it in your favorite video player after loading the video.
For example, here's the results page for a search of Korean subtitles to the critically-acclaimed 2009 film "Bruno"
UPDATE - JAN 2014:
Here are some additional useful sites (for television and/or movies):
- Subtitles Archive (sample search)
- GomTV (sample search)
- Jamak Nara (sample search)
- nScreen (sorry, can't copy the search URL directly)
Once you've got the subtitle files, check this post if you have any issues with encoding.