The fishing boat overturned just before midnight on Saturday, with three crew members picked up by rescuers, the coast guard said.
Video released by the coast guard showed a rescue boat approaching the overturned fishing vessel, with the three crew members in the water near the red hull.
Coast guard ships were scouring the area for nine others, and President Yoon Suk-yeol ordered other naval and civil resources to help, his office said in a statement.
In late January, at least eight people were reported dead after a cargo ship sank between South Korea and Japan, Chinese officials said.
(Reporting by Josh Smith; Editing by Christopher Cushing)
