There's really no need to store banana bread in the fridge. If you're going to eat it right away, it's perfectly fine to leave it on the counter. If you plan to keep it for a few days, it's best to keep it sealed in an airtight container at room temperature. Banana bread kept at room temperature will stay fresh for about four days.
The best place to store the banana bread that you plan to eat right away is on the counter. Never store it in the refrigerator. A refrigerator circulates cold air, which eliminates moisture and can prematurely dry breads and pastries. If you can't wait for frozen banana bread to thaw on the counter, or if you crave the sweet smell of bananas and sugar that spreads around your house, defrost the banana bread in the oven.