If there is an actor I like I'm more willing to give it a go, or if it's based of a story I already like (like To The Beautiful You), but mostly I base it off how interesting the synopsis is when I read it. If it's a historical, I'll turn away almost 100% (they're just not my thing) so I tend to give the genres a check, but you can usually tell if it's a historical in the synopsis. Then it's the first episode that judges if I will continue watching it. 
1- The storyline
2- popularity (sometimes)
3- the number of episode must be 21 or less than that
4- actor
the actor is last bc i will always end up loving them after watching their acting hahah but sometimes if i hesitated to watch, i will read through the comment and ask them is it worth watching them and such and i will try watch the drama on my first 15 minutes and decide ;)
1.Synopsis/ picture. (the picture catches my eye first then I read the synopsis)
2.Rating
3.Comments (if it has aired.)
4.Actors (3 & 4 are switched sometimes)
5.Length (If the drama is long...well I might pass.)
Length doesn't matter for me. If the drama is popular and everyone's talking about it, I'll give it a try. I think the most important criteria for me would be the storyline/plot first, based on the synopsis. If it's about something I have no interest in, why would I bother watching it? Unless, and here's criteria #2, the cast is full of actors I admire and love (such as Ha Ji Won, Jo In Sung, etc.) Besides that, I'm good. The easy thing for me is to start a drama, but it's harder for me to stay dedicated, especially if the acting or production is flawed. 
Story, ratings, length (episode #), and actors
 for me it's mostly story/ genre. there is no actor-even ones that i adore- that could make me watch a drama that doesn't interest me. no amount of So Ji Sub can make me want to watch Ghost, sorry :/ there are a couple of actors i really try to steer clear of- kim tae hee for example- but i will watch something even if it sounds fun even if  i don't know any of the actors.
I usually choose a k-drama based on who the writer is. I'll check a drama out if it has an actor I like is in it. 
1- Unfortunately, I'm the kind who judges a book by it's cover, so the picture and the title need to catch my attention, they are the first things I'm gonna see so they need to catch my attention otherwise I won't even get to take a look at it, but I might still watch you with an ugly picture or bad title if I feel like, one never knows... Which is a reason to why I end up only watching k-dramas and rarely watching dramas from other countries... 
2- Length 
3- Genre 
4- If I can find the drama with good subtitles and quality 
5- Actors 
6- Ratings 
7- Reviews
8- My mood  
9- I like to read some spoilers too, if it has a good ending, if someone dies, if the characters are likable... 
10- I don't like to read synopses, most synopses just don't make justice to the dramas or movies, so I avoid them, but when I reeeeeeeeeaaaaally can't find anything to watch I do read them... 

Yeah, I'm picky, that's why I have so many problems choosing new dramas to watch and end up watching mostly airing dramas... 
A combination of factors, I guess.

- Plot is important because if I don't like the story I would think it a waste of time to watch it. 

- Then if there are actors I like, I'm often willing to give it a try even if the synopsis doesn't sound very interesting to me. I usually give a drama 2-4 episodes before I decide to drop it, if that's how I feel about it, but sometimes I can barely sit through one ep. 

- Recommendations. I don't consider ratings/popularity, but I read reviews sometimes. If someone with an often similar taste recommends a drama, or someone who knows what I like, then there's a fair chance that I'll like it. 

- Length. K-dramas are usually longer than j-dramas, and if the plot or actor/plot combination doesn't seem appealing enough to me, I'm hesitant to pick up a drama longer than 16 eps. Often I find k-dramas more draggy with more filler than j-dramas... but of course this is not always the case! (And I'm getting used to watching longer dramas!)

- My mood. It depens on what I feel like watching at the moment. 

- Genre. Generally, I try to avoid some genres.

1. storyline is most important, has to be able to keep my interest as a whole
2.dialogue..the writing has to be able to stay on par with the storyline
3.cast.. a decent cast that fits the roles not just celebrities to try and sell a lame duck
4. review, while i appreciate input... as is beauty in the eye of the beholder so is my interest in what i see, read or hear..

genre fits at the top since that is what eyeballs me into reading they synopsis in the first place
plot
reviews
and recommendations (sometimes I'll watch something just because someone won't shut up about it. 9 times out of 10 though i'm glad i listened. except for Boss and Me. I should not have given in to that one.)
The synopsis and the year it was made. I honestly don't like dramas before 2009 and even dramas before 2012. It really depends, but a lot of times I get into dramas if I see one of my favorite actors is in it.
I usually watch the first episode of whatever is featured on viki. If I like the pacing, then I'll keep watching. I always drop when I find that the episode goes tooooo sloowwwwwwww. So the most important thing: pacing. 
I choose a drama based on the plot, but sometimes I can watch it even though the plot sounds meh if the poster or soundtrack really appeals to me.