Google says...
Ripening your Banana
The bananas that are shipped to many parts of the world (from places such as Colombia, Costa Rica, Ecuador, Guatemala, Honduras, Panama, Mexico, and Nicaragua) are picked green from the trees, and are kept from ripening until they reach the shops. As the banana ripens, all that mealy, squishy starch is supposed to turn into sugar. Unfortunately, if the bananas are refrigerated at any time between being picked and being eaten, they will not ripen properly. So don't stick them in the fridge. Imperfectly ripened bananas are composed of starch; but as the natural ripening proceeds, the saccharine material is converted into dextrin and glucose. The best thing to do is to get a paper bag and store them somewhere at room temperature. If you can, find a way to hang the bananas from a hook, you'll avoid bruising the bottoms. When they look ripe enough, stick them in a fridge (again, in a container that protects the bottoms). The skin will blacken, but the fruit will remain at the same ripeness.