I have three basjoo's in pots and they are still outdoors unprotected here in sunny Brum, I wait until the leaves look brown and tatty (normally after first frost December) then chop em off just leaving the growing tip leaf, this just makes it easy for storage.
If it has to stay outside, remove leaves, buy a piece of plastic netting, roll it into a tube and tie to secure, then cover the tube in several layers of fleece and tie top and bottom to keep in place then simply pop it on the leafless pseudostem when the weather is freezing, remove on fair weather days to eliminate rot, a thick mulch may improve its chances too. I call this my “Musa Matey” ;D