I’ve found two terrific—and simple and cheap—solutions.
One, I call a blast gate and the other is, get this, a piece of cardboard.
Old House Remodeling: Blast Gate
This name may be ambitious, but the technique delivers.
To seal one of the two entries to this project (a kitchen in this case) I fabricate what amounts to a screen door frame. It fits an opening in a passageway, in this case the entrance to the kitchen hallway. I cut 1×4s to fit inside the opening with at least ¼ inch of clearance all around. I join them with drywall screws a ½ plywood gussets.