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. Yes, you can leave the banana bread out overnight to cool if you want.
However, it's best to store banana bread in an airtight container. 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.
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.