Try Quake Hold - usually used in California and other shaky earthquaky places to hold figurines on shelves so they don't go flying. http://www.quakehold.com/safety-during-earthquakes-safety-preparedness-safety-furniture-straps/safety-preparedness-safety-furniture-straps-collectibles.html

Beautiful cart! I would try first to adjust the legs/feet to balance it out on the floor....or is another part of the room more level for it? Felt might work, too, but I would level it off first.

OR would you have something pretty....and heavy......to display in the ice chest while it is not being used? Colorful glass orbs?

http://www.thefind.com/office/info-self-sticking-felt

also you could get a piece of carpet with backing for under the cart and add a piece of plywood under it to keep the cart leveled.