Compatibility summary
You will need a travel adapter in Saint Kitts and Nevis; plug types used there differ from South Korea.
Plug comparison
South Korea plug types: C F
Saint Kitts and Nevis plug types: A B G
There is no common plug type between South Korea (C F) and Saint Kitts and Nevis (A B G). You need a travel adapter.
Voltage comparison
South Korea: 220V, 60Hz
Saint Kitts and Nevis: 230V, 60Hz
Voltage difference is within a range many devices tolerate; check your appliance label.