South Korea will very likely not take action against North Korea unless South Korea is invaded or severely threatened. South Korea certainly will not take action against North Korea on its own. If action were taken, it would be done with other countries involved. These countries would include the United States and Great Britain. The United Nations would very likely have a role in some action.
There are some scenarios where the action might be taken. If North Korea launched a missile at South Korea, South Korea would respond. If North Korea invaded South Korea, there would also be a response. There currently is a lot of discussion about taking action against North Korea if North Korea develops a nuclear weapon. While there may be other scenarios involved regarding taking action against North Korea, before any action was taken against North Korea, South Korea would consult with the United States as well as other countries and other organizations. Since the Korean War ended, there have been opportunities to consider military action, but no major actions have occurred. Most incidents that have occurred have been near the border of these two countries.