I've been getting these questions a lot lately. Here are my answers.
1) Work for a reliable institution. You are usually safer with universities or public schools (or programs working with public schools). As a general rule of thumb, the larger the institution and the older the English teaching program, the safer it is as a workplace. Here are some I recommend:
2) Start early. Korea requires more and more documentation and authentication every year. Some of those documents can take a great deal of time to obtain.
3) Maintain healthy skepticism. Most employers and most recruiters are pretty good people who will treat you pretty well, but there are some real scumbags out there. Check the wages compared to other jobs with similar working hours. Ask to speak with a current or former employee (a foreigner if possible).
4) Check out the area. While public transportation here is good, its quality diminishes dramatically outside major cities. Be aware that if you live in the sticks, you may have to buy a car or suffer the randomness of buses beyond your control. If you're a Christian, you'll want to be looking for an English speaking church within reasonable commuting distance.
5) Know your rights. We're currently working on an English Teacher's Rights guide. Stay tuned here for that posting coming soon. In the meantime, make sure your potential employer plans to provide you with health insurance and pension.