First of all I look at the poster because it must be the first thing to catch my attencion. After that I read the synopsis and then I watch some episodes.
Look at poster and synopsis. Then, I look at the trailer if it have one there...if not, search online. If it looks good, I watch a few episodes. If it is my thing, I continue; if not, I drop.
Actors, Synopsis, and watch the first two episodes. I never trust posters or trailers, 99% of the time they are misleading and don't do a show justice.
1. Actor/Actress 2. Plot 3. Length 4. Genre 5. Depending on my mood I'll go on a drama binge of dramas from just 1 country. (I was totally into Chinese wuxias for a solid 4 months :-)
For me the first thing to do is reading a synopsis, then if the promoting images are atractive i give it a chance. The actors are not so important to me but i would see a drama more enthusiastically if there are actors/actresses i like.
Various reasons for picking which dramas to watch: story, cast, genre …. One very specific reason for NOT to watch is if an actor/actress who cannot act is in it.
Mainly genre/theme that matches my mood, like: struggles in Joseon Era; single girl in a apartment with a roommate; detective/medical; tragic love loss; owning and managing a bar/restaurant; life in high school as a teen; etc. But I'm heavily influenced by main actors too. I also give in for the most popular dramas that get a lot of passionate reviews.