Yes, the big stems that you cut off go to mush when the weather gets cold anyway, David. Bananas are monocots (only one growing tip) so once you cut a growing stem, it dies. There's no harm in leaving some of the stem tissue to nourish the roots but you can cut it off wherever, and it really doesn't make any difference. You can chop up the cut off stem and use it as part of the mulch, too. It's rich in all the right nutrients. (if you're feeling really energetic, or have a chipper/shredder)
Down here, when bananas have borne fruit, the stem that fruited is cut off at the base to make room for new shoots. Even a stem that is 8in. in diameter can be cut through with a bread knife. They are not woody like tree trunks, they are fleshy stems. If we do get a run of really cold frosty nights the bananas are a very sad sight, just piles of collapsed stems and brown leaves. Freezing bursts the water-filled cells of the stem and they are so top heavy they fall right down in one night.
Was going to suggest you might want to separate off the pup on your big plant, pot it up and bring it into the basement, in case it might not survive the winter. Don't cut (or break) it off though, if you do it won't grow next spring.
