How should I clean, or care for, a banneton (brotform, proofing basket)?
A stiff brush and a set out in bright sun is about all I've felt the need to do with mine.  


Normally I'll just tap out any extra flour.
If I've got a dough that sticks (like last night when I didn't flour enough), then I just take a stiff brush and get all the bits off.  
If they do get a bit wet, dry them with a towel and then in a warm oven.
If I think there's been enough moisture to be concerned about something growing, I'll set it in bright sun (or my UV exposure box).  Strong UV is a good inhibitor for much of that.