The Korea of today is as healthy and hygienic as any industrialized country. Although there are the occasional outbreaks, Korea very rarely in the past thirty years has had a contagious outbreak of anything. The vast majority of Koreans get their vaccinations done, or at least as much as any other country with its medical facilities available.
One thing of note would be the outbreak of MERS, or Middle Eastern Respiratory Syndrome, that occurred in Korea in 2015. The outbreak was traced back to a man who had recently traveled to Saudia Arabia and unknowingly returned to Korea where it rapidly spread across Korea. There were over 180 cases of MERS. It included a death toll of 36. The Korean government responded with quarantine of known patients and guidelines on how to prevent the spread like wearing facemasks and an increased washing of hands. Hospitals, schools, restaurants, and other public places made sure to implement these kinds of guidelines. The Korean public eagerly took up these practices as it was difficult to figure exactly how the disease was spreading from person to person. The outbreak lasted almost two months before being successfully tamped down.