I tend to drop dramas when i find it not really suitable for my liking or the plot isn't engaging enough (even i only see the first ep)
But in some cases, if the drama has my fave actors/actress in it (at least one!), i will finish it no matter how boring it is.......(i am a really dedicated fan, aren't i? XD)
I don't like critiquing so I see no point in finishing something I don't like so will definitely drop it.  
I initially dislike dropping dramas because I don't like the thought that I have unfinished dramas. But I'm slowly learning dropping dramas. Reasons are either 1) they those which I cannot tolerate anymore or 2) ones I've put on hold before but after some mulling over realized that I don't care about anymore. But I can finish them too. Just so to give some a chance or because I'm planning on blogging about them. (But I usually speed-watch when it comes to that.)
well i never like to consider dropping a series i only put it on hold for a long time, or until i find it mood and time to continue watching it.
i am still new to the drama world so i have yet to find an such series that i wanted to put on hold yet.
I usually do finish the drama no matter how boring I find it. The only reason why I'd drop a drama is if I can't find videos to watch it anymore or if I don't have time to really watch it anymore. The most current one I had to force myself to finish was "tomorrow With you" I had to keep skipping it but I finished it.  There's no way I'd actually watch 16 hrs of it if I hated it.  
I often wait till episode 5 or 6 to decide wether I want to drop it or not if I don't like the plot from the beginning..
For me the bigger problem is when there's one character or one thing that really bothers me, but I love the rest of the drama. That makes it hard for me to decide wether to drop it or not. If I think a plot is boring I'm dropping it immediately, because I don't want to drag it further... But sometimes I'm picking up dramas again, because I wasn't the right time yet..
I don't like 80% of the dramas I watch because I like torturing myself :(
Most of the time....no. But once in a while I do, but finishing it is painful lol.
If I'm watching with someone else, it's easier to finish it. Usually I drop dramas though. I recently dropped a drama on the first episode lol 
I usually skip or fast forward those boring parts and if it is still boring I usually end up watching the last episode
I don't like to drop dramas because I want to finish what I started. But If I find something unwatchable I just delete it from my list and I think of it as if I never watched it (for example On the Way to the Airport, I couldn't stand it, I watched just an episode, and then I dropped it). But if I find dramas that in some ways are interesting, but at the same time are boring at moment, I just keep watching (if I couldn't stand them, I just start another thing, and sometimes watch a couple of episodes of them). For example The Journey of Flower, I found it strange, and I couldn't understand well the story or the characters. After about 20 episodes - I don't remember well when - I started to love it with all my soul. The same for Six Flying Dragons, at first I didn't understand the story - because I didn't know Korean history - then after 15 episodes I started to love it. However there are cases such as for the K2 o Faith - in which I found very hard to finish them, but in the end I finished them without love or like them neither a while. Everyone has his/her personal tastes. Now I know what I like and what not, so now I do start only what I think I might like :D
Depends I do if the first ep wasn't good too me, like the plot isn't as interesting as I thought. Yet I don't if I really love the cast I tend to think that it'll be better as time goes by. 
If I don't like a drama,I usually drop it,then months down the line watch it again....I feel like I might have a change of heart,but normally even then I still can't force myself to rewatch so I drop it for good...DOTS,K2,and W are a few I just couldn't enjoy no matter how much I tried to rewatch it-.-   
for me it's depends on the lead. If the lead is/are my bias then i'll force myself to finish the drama.