To keep the bread very moist in the fridge, you'll need to add an extra layer of protection. I suggest loosely wrapping the cooled loaf of banana bread in a piece of parchment paper (to prevent sticking) and then wrapping it tightly in plastic wrap. Place a paper towel on the bottom of an airtight storage container OR in an airtight bag. Then place the cooled bread on top.
The paper towel will absorb any excess moisture that may seep over the next few days. Don't let anything stand between you and your banana bread, avoid these common mistakes in banana bread. Moisture is key when it comes to banana bread, and the ratio of flour to banana makes all the difference.