can you post a picture ?

bananas are a favorite for spider mites. My guess is you have a bad invasion of spider mites. The white cottony stuff is their web. I am not endorsing a product but I use Shultz Garden Spray (Houseplant and Garden Vartiety) to control bugs if my plants have a bug problem. You can buy it at the big box stores or Walmart.

To prevent infestations, when you are at the store of your choice, buy a clear clean plastic spray bottle and fill it with water and spray/mist your indoor bananas (for that matter all your indoor humidity loving plants) daily or every couple of days. I write "water only" in magic marker on my spray bottle so no one puts anything else in it. It is also a good idea to run a fan in the room with your plants for air circulation and this too will help with bug infestations. Run the fan for a couple of hours every night when you get home from work. It does not have to run all the time but 4-8 hours a day would be ideal.

Go get those bugs!