As long as they don't get attacked by spider mites they should stay healthy but dorment until next spring. I have been overwintering my bananas in conditions similar to yours for several years and they all always stay green but don't grow. Good luck!

how do I prevent spider mites? I have "fruit tree spray" that contains pyrethrins - should I use that? thanks

I overwintered a dwarf cavendish pup last winter and plant to bring the plant in next weekend.

You may not need any sprays but what will help alleviate the spider mite problem is to try to keep the air humid around the plant. I had mine in a cool bedroom literally right next to my room humidifier. The room got no direct sun in winter but was moderately bright during the day. I majorily reduced the watering - basically only enough to keep it from going bone dry. Alternately, if there is a way to put the plant on a pebble tray with water, that might also help (although that may be dependent on how big the plant is and whether enough evaporation from a pebble tray would even reach up and around the pseudostem and remaining leaves).

I have a banana tree that dug up and overwintered last year, and it grew one leaf every 2 months, and when every leaf came out the rest just died off. I watered it about once a month until it started to grow good again.

The best way to avoid spider mites is to make sure you don't bring any into the house to start with.
Spray your plant thoroughly to the point of runoff on all leaf and stem surfaces with a good insecticide for gardens and let it dry before bringing it in. Then when it's inside, keep a good eye on it and make sure that none are attacking, take it to a heated garage or somewhere safe to spray and spray it again at the first sign of mites.
Don't overwater your plant inside, feel the soil and let it dry out between waterings, then water it thoroughly but don't let it stand in water, and let it dry again before watering.
Put it as close to a window as possible and turn it from time to time to make sure it doesn't grow all one sided.
Use a water soluble fertilizer 20-20-20 at 1t per gallon while it's inside.
Don't ever pack potting soil around the roots, let it settle in loosely, use a good potting soil like promix hp, or add perlite 40%/60 potting soil to increase drainage.
This is my best advice.
Sandy @ Northern Tropics

keep it dry, and trim leaves when they yellow. south window will help too.

Here is a link that might be useful: banana growing info