Taking birth-control pills to South Korea

Having a prescription or not does not matter. Many countries, among them South Korea, only allows you to bring medication for personal use for a period of three months when arriving from abroad. If you are staying longer, you will have to stock up your supply there and potentially get a prescription if required.

Birth control pills seem to be over the counter in South Korea as well, so you actually only have to go to any pharmacy and buy your pills there. You should perhaps check if pills with the same active ingredient(s) as you use at home is available in South Korea and if not, you could anyway in advance try to find a suitable substitute. If you are coming from the US, birth control pills are probably by a ridiculous amount cheaper in South Korea than in the US, so the most economic solution is probably to just bring enough for the first few days and then get everything you need after you arrive.

From someone who has lived in South Korea their entire life, and flown in and out regularly:

Birth control pills are over the counter, so no prescription is needed. They are available in pretty much every pharmacy, and there are plenty of pharmacies in every major city, open at every hour of the day. The pills are somewhat expensive, but not prohibitively so; they generally cost around 24000KRW (~20USD or ~18EUR) for a 1 month strip.

You may of course want to bring some pills just to cover your first few days or weeks after arrival. To be on the safe side, you could bring the original container and your prescription. A translated prescription seems excessive, and it is highly unlikely that these pills will cause any issue at all at customs.