As a last resort it would be necessary if they want to continue to exist as South Korea and not just a new country, but there's other ways to simply incentivize procreation, which they're not doing enough of.
They really have 3 final choices, assuming they either don't or unsuccessfully try to incentivize consensual procreation:
1) Let the country collapse with its dignity
2) Mass importation of foreigners, reshaping South Korea
3) Adopt a "childbearing draft"
In order of likelihood, it's 2, 1, 3. Modern, civilized people would rather have their life upended or just plain ended than be called a mean word like racist or sexist, so they will undoubtedly drown themselves in immigrants or maybe go down with the ship (which would be interesting to watch as a case study and warning nobody will heed).