My one story home, located in Columbia, South Carolina, was built in 1937. I purchased it in October of this year. 

There is a stairwell in the center of the house. In the hallway at either end are doors. One stairway goes to the small concrete pad that originally contained the furnace, and the other leads up to the unfinished attic. Both stairs have gaps between the headers and the risers. 

As it was designed, I think that on warm nights, the attic door would have been left open to draw-in cooler night air through the screen windows, and that on cold nights, the basement door would have been opened to draw extra heat from the furnace. 

Modifications have been made since the house was built. There is no longer a furnace. There is now a heat pump on the side of the house.  There are now storm windows, and the heat pump provides air conditioning in the summer. Soffit vents have been installed in the eaves. The highest part of the roof has been modified by the addition of a roof vent. Insulation has been put down in the attic. Insulation has been put on the sides of the stairwell. Some parts of the subfloor above the crawl space have insulation. 

The crawl space around the small concrete pad is vented on all sides of the house. This week plastic was put down over the entirety of the crawl space.

There is a constant breeze in the stairwell. The insulation was put in with the kraft paper toward the stairs, and it has to be replaced. 

Because the stairway would act as a chimney should the house catch fire, I am concerned about replacing the stairway insulation with anything that is flammable; as I understand it, that leaves me with choosing between rock wool and fiberglass if I go to a big box store. Any other ideas?

Going forward, should I seal off the three sections of the stairway (crawlspace, main floor doors, attic)? And if so, what of these has priority?