Nah. Mine have a top like that. (Arkel Shopper)

It depends on what you buy. Rain doesn't bother produce, frozen, meats, deli, canned goods, or bread (in plastic). Even snack food like chips are fine, and I can leave then sticking out the top so they're not broken, even in the rain. It can be a problem for things wrapped in paper or boxboard like flour, sugar, and breakfast cereal.

In any event, I've had mine for over four years now, without ever damaging anything due to rain.

If the open top really, really bothers you, then there's the Utility Basket which closes completely. Makes it a problem to leave the celery sticking out the top, though.