Any decision about refrigeration or non-refrigeration of bananas should depend on their degree of ripeness. If a banana is not yet ripe, refrigeration will cause chill injury and disrupt the ripening process. If an unripe banana is refrigerated, even if it is brought back to room temperature, it will not ripen. However, if you have bananas that are already ripened, it is fine to keep them in the refrigerator for a period of several days. You're likely to see a darkening of the peel, but the inner flesh of the banana itself should not become discolored or harmed in any way.