My answer has spoilers regarding this storyline, just as a warning for anyone reading this.
To add to what Kat said, that particular storyline is done quite tastefully from both sides of the spectrum I think. The character plays a major supporting role (as the male lead's best friend) but the homosexuality theme is a subplot and does not play a major role in the overall story. As Kat mentioned, the character (Jun Hee) has a crush on his friend (Yoon Jae) which he discloses to the female lead. The focus is on the character trying to figure out who he is, so the subplot is in no way in your face or anything like that (and the storyline doesn't even come into play until a good way into the show and is in the background). There is never anything shown on screen and even the ending for the character is left fairly open (it is implied but never shown or explicitly addressed), and I consider it to be quite respectful given the conservative nature of Korea, while still keeping the character's dignity in place and treating the story with tact and significance. Basically I am saying that I think it should be fine. My parents are very conservative and I would be fine watching this series with them so I think it should be completely acceptable for your setting, though when the episode which opens up this topic comes up you may want to have a a conversation about it with your students.
