Replacing an exterior screen door footer threshold to close gap
Would installing a pressure treated 2x4 across the door frame solve
this problem? Or is that a short term solution as it would rot out
from the water/moisture that gathers in the stairwell?

Yes it would, you may have to do some creative cutting to get it fit well. It is treated so although water may gather there it will not be a short term solution and will last for quite some time.
If you install it with removable fasteners then in X years, however long it takes to no longer be sufficient,  you can just replace it with another piece of treated.
I would coat the wood with truck bed liner to make it last much much longer. Truck bed liner is available in a quart can, it can be brushed or rolled on or a spray can, it  will be tough and hold up to foot traffic.