I did when I first started watching dramas, but now I quit watching if I get too bored with it or dislike it, or if I learn the ending is something I'd hate.
I rarely drop dramas, only if it's unbearably bad, like...Eyes are bleeding - bad X_x which does happen, but not often. What I tend to do when I come across a drama I don't like is that I speed Watch and finish it that way because I want to know how it ends.
Watching drama is like being in a relationship; if you dont have any reasons to stay then stop. HAHAHA
Sometimes I do and sometimes I don't. 
I'm someone who drops dramas I actually like, so yes, if I'm not like a drama then down it goes. 
I personally really don't like dropping dramas (slight OCD?). I do drama reviews and like to give honest opinions, especially if I'm disliking a hyped drama. I also feel as if finishing dramas (even the bad ones) gives me a better idea of what's good and bad in the drama world and helps me form more solid and constructive opinions. But I don't like to waste my time so if a drama is really that bad I speed watch (skip all the flashbacks, dramatic staring, etc.) to be more efficient. I also usually look at reviews and other opinions before I start so I tend to watch dramas the I enjoy. :)
not usually but i recently finished "Second Love"  worst drama ever.  I also completed Missing You which almost flooded my house and drowned me with the amount of crying on the screen - it didn't make me feel sympathy, it just made me want to slap them all.  I also completed DoTS..what a waste of my time.

I thought i learned my lesson with MY and DoTS but somehow convinced myself that since SL was only 7 eps long i should finish it.  I won't do that again
Life is too short for bad dramas.
I practically drop only one drama until then. When I watch, I watch until the end, being bad or not.
i won't feel satisfied until i've completed what i started, so that makes me sit through dramas that i find boring or absolutely hate.
but recently i've started dropping dramas that do not interest me. 
Sometimes I finish the drama other times i stop watching.
I have a 3 episode rule! If I still really don't like it after 3 episodes, then I generally give up. 

However, I know there have been times that I stuck with a drama I didn't necessarily love because I had hope that it would turn out better. Or something in the drama happens that piques my interest, and even though 80% of it is boring me, and I'm on my phone, I'm waiting to see if that other 20% will wow me. And that's how I sometimes end up finishing a drama and then going, "Why did I watch this?"

But if you look at my drama reviews, you won't ever see anything rated below a 6. That's because I don't bother to finish dramas that would be a lower rating to me! ^.^

Drop it immediately after 1 episode if its too boring. 1 hour is a really long time to be watching something that doesn't interest you. 
But for anime its different always finish every anime because sometimes i underestimate an anime and it just gets better with each episode and eventually becomes one of your favourites and each episode is only 20 minutes.  
But this is just my thing that i do xx
I just can't drop a drama once I start it, even if it's really bad. I just keep it on hold and come back to it later (as in years later)
i don't like to drop dramas, but I am good about watching an episode before adding it to my list if I'm not sure about it.  I wouldn't say that I've ever finished one I didn't like but there have been quite a few that lost my interest along the way and I ended up forcing myself to finish later on.  In some cases where it's beyond repair and extremely predictable I will cheat and find a summary of each episode and watch the last 2.