I recommend G&M Clinic in Munhwa-dong.

This is the place I visited because of skin troubles. The nurses at the reception desk were also kind. The doctor was more straightforward than overly friendly, just stating the facts, which made it easy to understand, so I liked it. Although the Google rating is low, I was okay with it. What I wanted to know was how they would prescribe the medication and what treatment would be good for me. I was glad that they answered my questions.