Insulation, Radiant and Vapor Barriers - Stairs exposed to crawlspace

I have a split level home with crawl space under the living room which is only room at entry level. My the underside of my stairs going to lower level are fully exposed to crawlspace. Is it normal for stairs _not_ to be insulated from crawl space? The rest of the interior walls in crawlspace are insulated. I stapled and taped foil insulation to seal off the stairs (two regular sized gaps between three studs) , but wondering if I have done something to upset the "system"? Is there a better approach?

Also, I'm seriously thinking about encapsulating my crawlspace using 20mil on the already stone covered floor (has plastic underneath, too), sealing vents and insulating the exterior walls. I only have two concrete exterior walls, and two outside vents on one of those walls. Do I need to do anything special with the existing interior insulated studed walls (and stairs situation I mention above)?

You should absolutely insulate the underside of those stairs. You may need to frame 2x4's down from the stringer bottom edge prior to insulation.

You should not cover up those vents. They are required to remove moisture from the crawl space. You may want to purchase some white insulation foil/paper (comes in 4' rolls) to cover up the under side joist insulation and the knee walls.

An additional layer of plastic over the washed gravel is not necessary if you have a vapor barrier under.

hope this helps, jim