Never used a cover or wiped down the cart. I think maybe those anti-bacterial wipes they have near the carts scare me more than any germs? I don't want to touch them and then have to go wash my hands.
4 kids who have always sat in the cart and they are almost never sick... and when they are it consists of a low grade fever and they sleep a bit more for a day or two, that's it.
If I had a kid who was always coming down with something, I might do things differently. Like, put them in a baby carrier every single time, instead of use the seat in the cart.