Yes, some Korean people do eat dog meat. There are a couple traditional dishes made out of dog meat (the most common is boshintang, a stew). Dog is also eaten in other parts of East and South Asia, including certain regions of China and the Phillipines.
There are some records of people eating it thousands of years ago in Korea, when the country was extremely impoverished. However, it is not a regular part of Korean cuisine. I've never seen it served in someone's home, for example, but I have seen it advertised on signs outside a specialty “dog meat” restaurant in Korea. It seems to be popular among a certain group of older gentlemen for its supposed power to enhance stamina and virility.
Even though a fair number of Koreans (anywhere from 5 to 30% depending upon whom you ask) may have tried it before, it is only a very small percentage of the population that eats it regularly. There is a large and vocal group of Korean people that are against the practice of eating dog meat, and want the South Korean government to enforce the laws in place that made dog meat illegal. There is also a large population of people in South Korea that don't eat or enjoy the meat, but do feel strongly that it is the right of others to do so. There's a smaller but still vocal group of pro-dog cuisine people in South Korea who want to popularize the consumption of dog in Korea and the rest of the world. So even Koreans are divided on the issue.