There is no dress code. In general, most schools are ok for teachers to wear casual clothes if they are clean and decent clothes but it is always safe and good to wear semi-formal clothes.
For male teachers: collared shirts, dress shirts and sweaters with a nice pair of slacks would be ideal. Some schools may ask you to wear a tie, however by no means are they required through.
For female teachers: collared blouses, dress blouses and sweaters with a pair of slacks or a nice long (knee-length) skirt would be the best.
Mini-skirts, ripped or torn jeans or t-shirts with offensive slogans, as with teachers in American schools, would be frowned upon and you would be sent home to change. Keep in mind that you are trying to dress in a manner which presents you as a serious teacher to your students, students' parents and business directors.
It may also be good idea to bring at least one business formal suit or outfit along to Korea, as you never know what occasions may pop up . The better prepared you are, the easier you’ll be able to dress appropriately for any situation.