Crawlspace & Mold

[Stavs]

I have a little storage room on the first floor of my home. The walls are drywalled and I have a mold problem. I started tearing out drywall on the back wall and I found that there is a crawlspace area behind it. There is no type of vapor barrier or insultion in this room or on the walls. When I go to replace the drywall in there, how do I keep the moisture from that crawlspace area from damaging my drywall and letting mold into the room? Can I just put up a sheet of plastic between the crawl space area and the new drywall? Also, should I have that crawlspace insulated (I was thinking spray foam type)? It sits below my kitchen and our floor is always cold. My main goal is to keep moisture out of the crawlspace and the room. Any help is greatly appreciated.

Thanks,

Stavs