I do indeed download dramas, but I use a torrent so it’s not a straight download. You’ll need a bittorrent client to use the sites I mention.
Avistaz is probably the most reliable and easiest to use if you have an account there, although they do have strict leeching rules (but I think completed shows are usually “free” and don’t count against you). English is the website default so it’s easy to find things there. They’ll also include a download link for any subs available for that show.
A popular Korean torrent is torrentkim – if you google “torrentkim” you’ll find the most current one (I think it’s up to number 10) which is reliable but hit-or-miss if you’ll end up downloading an episode with the annoying sex ad on it (SO annoying, even if it’s only text that shows up a couple of times, and is why I’ll head to avistaz if I need to make sure a download is free from the ad for screenshot purposes). It’s all in Korean so you’ll need to brush up on your hangul, or if nothing else copy/paste the hangul drama name into the search.
There are other Korean torrent sites I’ll use if need be but torrentkim is the biggest so it generally has more seeders, which makes things faster/easier to download. Torrent sites can randomly disappear/change for obvious reasons, so it’s tough to suggest specific ones.
If you’re looking for subs, subscene will have whatever you’re looking for if it’s available. Obviously these are softsubs that you’ll need to sync to the show you’ve downloaded.
The Obsessive Dilettante 