Heat rises up stairwells. One reason to have a door on a stairwell would be to keep the heat (from a fireplace or a stove) on the ground floor, instead of rising up the stairwell. This would keep the ground floor warmer.

This would be particularly useful if the stairwell went up from one of the main rooms in the house, rather than from some separate hallway (which could be closed off from the main rooms by doors).

Would you be willing to post picture or an example? I'm afraid I'm not that familiar with the vernacular architecture of the American South.