Damsel in distress is used as a phrase to mean a woman who needs to be rescued by a man. Damsel is a woman with victim mentality.
There isn't a perfect phrase in Korean that holds meaning & a broader tone of "Damsel in distress".
Some phrases that I can think of are below.
- 위기에 빠진 여성
- 붙잡힌 아가씨
- 곤경에 빠진 그녀
You would almost rarely use or hear these phrases in real-life conversations similar to how you seldom use "Damsel in distress", but they can be sometimes found in writings, literature, books, or movies.