I think if mice want to be in they will be whether you cover them up or not.
My dad stores large stacks of maybe 400 bales under an open fronted dutch barn, no covers, the big round straw bales get stored outdoors too, but even though they're stored all year it's only the first 10cm that really go rotten and goes straight on the muck pile, the rest is good.
If you're storing in a shed as long as it's well ventilated you should be ok just stacking them up. It's if there's a condensation build up that you might find the outers of the bales going a bit fusty. When the bales have gone mouldy on the inside it means that the grass was too wet when it was initially baled up, so nothing to do with how you store them.
I wouldn't really cover them unless you have to store them outdoors, if you are then cover them as soon as they arrive, and maybe take the cover off on good dry sunny days.