If you are a nonresident service member, an honorably discharged veteran or the dependent of one, South Dakota treats you as a resident for tuition purposes, so you pay in-state rates at state schools.
There is nothing to apply for with the state. Tell the admissions or registrar office at your South Dakota school that you qualify for resident tuition status, and give them your service or discharge documentation.
Your military ID or orders if you are serving, your DD214 if you are a veteran, and proof of your relationship to the service member if you are a dependent.