In my experience I find it always better to err on the side of caution. This is definitely a question for someone of Korean descent, but on the surface my suggestion would be to at the very least make sure you're being respectful of the culture you're writing about. Learn what the stereotypes are so you can avoid them, maybe get guidance from someone you know that's Korean as well. I don't think there's anything wrong with writing characters from different cultures, but some cultures are more open/closed than others. What's important is that you respect the desires and needs of people of the culture in question.
Just as an example, in my story Cheri and Takir are both coded as being that world's equivalent to middle-eastern. They both face prejudice for their heritage, but I make it clear that the prejudice they're facing is unjust. Their culture isn't touched upon heavily in their current appearance, but I treat that culture with respect and avoid harmful stereotypes.
On the other end of the spectrum, there are some topics I avoid completely. One for example is a certain spirit that the Algonquin tribes of North America hold a deep respect for. I've had cases where others have wanted to use this spirit in writing, but the Algonquin tribes find even mentions of it to be very disrespectful, and have asked that the spirit not even be referenced by name. (It's name starts with a W and their beliefs suggest that saying its name in full is a way of summoning it.) As such, I avoid making any and all possible references to it.
Long story short, when in doubt, try to get the opinion of someone from that culture, and make sure you're careful of their beliefs especially spiritual ones.
