Popularity in Asian dramas is growing and many streaming sites are available to watch them on. I am going to briefly review each site, discussing the positive and negative aspects of each one. I am only going to be discussing the paid subscription sites but there are a lot of free Asian drama streaming and torrent sites.
Netflix
Monthly Price: $7.99
Netflix is the primary reason why I got into Asian dramas. The first drama that I watched, Boys Over Flowers, was on Netflix and I then proceeded to watch most of the the ones that were available at the time. I eventually moved on to Hulu and other sites because Netflix didn't have a large selection. It is a good place to start since Netflix has a lot of the most popular dramas from the past six years including Boys Over Flowers, Secret Garden, My Girlfriend is a Gumiho, My Princess, You're Beautiful, etc. I still use Netflix to watch movies (great selection of Foreign/Asian films) and American/British shows but I would look elsewhere if you really want a site mainly for Asian dramas.
Pros
Available on multiple devices
Many popular Korean dramas
Cons
Small selection of Korean dramas
Dramas are not available for long periods of time (they recently removed a large portion of dramas)
No other Asian dramas (Japanese, Taiwanese, etc)
Hulu Plus
Monthly Price: $7.99
Hulu is a great site and has an enormous amount of Korean dramas, both new and old. I first used the site to watch Coffee Prince and Wild Romance, it was my main source for drama watching until I found some better ones. I would like Hulu much more if it wasn't for the commercials. I do not know of any other sites that you have to pay to use that forces you to watch commercials. In all honesty, I would solely use Hulu if there weren't commercials because of their massive kdrama options.
Pros
100+ Korean dramas
Nice selection of new and old dramas
Available on multiple devices
Cons
Lots of commercials
Small selection of Japanese, Taiwanese and Chinese dramas
DramaFever
Monthly Price: $9.99
Unlike the aforementioned options, Drama Fever is a really great streaming site that is exclusively Asian dramas, movies and reality shows. They have over 100 dramas available and they sub newer dramas fairly quickly. Like Hulu and Viki, you can still use the site without paying but there is a longer wait for unsubscribed customers to watch subs and there are also commercials. DF also features articles, videos, and contests which are really fun.
Pros
Large selection of Kdramas
Decent selection of Chinese/Taiwanese
Available on multiple devices
Cons
Movies are listed as an episode instead of a movie
Poor selection of Japanese dramas
Viki
Monthly Price: $3.99
Viki is my favorite subscription streaming site. It has a lot of Asian dramas and moviesand also subs (adding English translations) new dramas within 24 hours. My favorite part of Viki is the drama commentary, usually added by subbers but now anybody can comment as well. It is a line of commentary at the top of the drama screen where people discuss or comment on actors/situations occurring in the drama. It is usually really funny and I will admit that it is the only reason that I made it through some overly draggy dramas. I do have to occasionally turn it off, as was the case when I watched Reply 1994 and the commentary just consisted of people arguing/insulting the two male leads. It is the cheapest subscription site although it was once free. The subbing is done by fans, subbers and segmenters do not have to pay for the site as well as exclusive access to certain features. You can use the site for free but you will have to watch some commercials.
Pros
200+ Korean dramas
100+ Taiwanese dramas
300+ Chinese dramas
Extremely fast subbing of new dramas
Dramas from Philippines, Thailand, India, Indonesia, Mexico, Vietnam, others
Large selection of new dramas and old dramas
Available on certain devices (Kindle, possibly Amazon Fire)
Cons
Doesn't always get the license for new and popular dramas
Poor selection of Japanese dramas
Kdrama/Soompi
Monthly Price: $6.95
Crunchyroll.com is mainly an anime site but they also had a Korean drama section. I was going to discuss that site until I found out that they made a sister site completely dedicated to kdramas. Kdrama.net is still pretty new but I signed up for a 30 day trial subscription to test out the site. It has a large collection of Korean dramas and will probably add more if the site is successful. I found a few dramas on KD that aren't available on any other streaming sites, paid or free. It has a lot of older dramas and some newer ones. While it does have some popular staples (Boys Over Flowers, Sam Soon), it is missing a lot of the ones commonly found on other sites. It has since been taken over my Soompi.com so it is now SoompiTV.
Pros
150+ Korean dramas and growing
Has older and harder to find dramas
Available on multiple devices
Cons
No dramas from other countries
Currently lacking some of the more popular dramas
Dramas listed under names different from other sites which can be confusing
There are a lot of options and the one you choose depends on your needs. Personally I would suggest Viki, Drama Fever, or SoompiTV as all three have a lot of content. Netflix is a good place to start and most people may already have an account. If you don't mind commercials, Hulu does have a nice selection.
*IMPORTANT* I am not affiliated with any of the aforementioned sites. I use/have used them either through paying a subscription or using a free trial. All opinions stated are mine and I tried to be as objective as possible.