There are several different websites where you can stream Korean dramas legally at no cost to yourself. Some of these websites even offer live streaming, so you can see the drama at the same time it is being aired on Korean television.
The most widely used websites for free and legal viewing of Korean dramas are Viki.com, Dramafever.com and Hancinema.net.
Viki is a community-based website that has English subtitles for its shows. It features Korean variety shows and movies in addition to dramas. However, because of copyright issues, some of the shows on the website only stream in the United States.
Dramafever is one of the most comprehensive Korean drama streaming databases. Having purchased copyrights from leading Korean broadcasting companies, including KBS, MBC and SBS, this website offers high quality videos and English subtitles for free. The website also includes Korean movies and music shows.
Hancinema.net is an independent database for Korean movies and shows, but it also features Korean entertainment news.
All of these websites have advertising to deal with if you watch them for free. You only get a commercial-free experience if you purchase the premium service for these sites. Free viewing also means you can't watch the dramas at the highest resolution available.