You sue an uninsured drunk driver by filing a Complaint at the courthouse. If you have uninsured motorist coverage that is easier to pursue than the uninsued drunk driver. Normally, people without insurance do not have assets. If you sue someone without assets you won’t see any money at the end of the day because they don’t have money to pay you with. Some lawyers take cases like these to get a big verdict. The problem is that a big verdict without money is just a piece of paper. If you have uninsured coverage than your insurance company will pay you and then try to pursue the uninsured drunk driver.