Hi Victoria! If the Korean health check is anything like the Chinese health check then you have nothing to worry about. Cold sores are a common and rather benign annoyance but nothing that would prevent you from getting a work visa in Korea, China, or any other country.
What most countries are really looking for in the health check is HIV. Most Asian countries view HIV and AIDS as 'foreign diseases'. A nurse friend of mine recommended using Zilactin for cold sores. She also said yogurt put directly on the cold sore is very effective too.
Let me know if you have any other questions. Good luck!