Grant is the closest inside the park to the South Entrance. However, we stay in Bridge Bay and have "easy" access to everything on the southern route.
If you main visit area will be geysers, suggest you try Madison.
Or, go just outside the park to Flagg Ranch. It is situated between Yellowstone and Tetons.
May be a bit more helpful if you explain what you are wanting to see in the area. Yellowstone is HUGE. Not really possible to stay at the south entrance yet be able to spend any amount of time on the north loop.