How do I insulate a crawl space under kitchen in basement?

I would not insulate the crawl space. If you have vents in the crawl, close or cover them for the winter. We have a crawl space under our whole house and it isn't insulated and it doesn't get cold. Heat rises. We have flood vents that get closed in the winter to keep the wind from blowing through. We don't have plastic in our crawl either. We live at the beach (on an island) in New Jersey and we have water come and go in our crawl. The water will percolate up and down and we don't want water to get trapped on top and cause mold. Some people here do have the plastic, but we've chosen not to.

You should insulate under the floor, between the joist, with the moisture barrier facing the heated room. Look into having the area spray foamed, it cost more but it is superior at sealing the space.

My kitchen and living room floors were cold being over the crawl space. I spent a whole weekend in my three foot high crawl space on my back. I insulated in between the joists with kraft faced fiberglass batts...paper side down stapled to the joists. Stuffed insulation in the rim joists. I also glued 2" foam board to the walls with construction adhesive. Don't cover the vents. You need air circulation to keep it dry. Can cover them for the winter from the outside. I also insulated duckwork that was in the crawl space with fiberglass wrap.

Go ahead and insulate under the floor, that will help keep your floors warmer. The moisture barrier always faces the heated side. Besides fiberglass batts or spray foam, there is also Rockwool insulation, which is fire resistant.