Get Out.
I've been bouncing in and out of Korea for the last ten years. There is a great deal of difference between what should be done and what is done.
You can go to Immigration and have your stay extended for circumstances such as flight cancelations.
Don't do it.
The wait may be incredible, and you may find that Immigration doesn't care that KE or OZ is on strike. It depends entirely on the mood of the official, it seems, whether you are quickly extended, or placed into the higher tiers of hell.
If your flight is canceled and the duration of stay is up, leave the country, cycle to Fukuoka in Japan, come back in, and you'll have either 90 or 30 days to get things straightened out.
Under no circumstances assume that because of a strike Immigration will understand the problem and ignore an overstay. There is a great push on now in Korea to solve the "immigrant" problem, with guest workers and illegal teachers heading the list. If you are on a 90 day visa and overstay, it will be assumed that you have been working illegally, and then the real fun starts.
I clear Korea Immigration on a weekly basis, and have found them to be amazingly nice, fast, courteous, but once something goes wrong, they tend to follow the rule book exactly. It's best to make sure that they never have a reason to look at you closely.
Don't take this as legal advice, merely based on the two hundred or so trips I've made into Korea.
M