If I buy a box of panko, I keep them in the cabinet. When I make my own, I keep them in a little baggie in the fridge.
Pantry, in an airtight tupperware thingy.
ETA: I keep homemade breadcrumbs in the freezer.
I don't think I've ever bought breadcrumbs, but if for some reason I make too many then I keep them in a zip lock bag in the freezer. I don't know why...