Refrigerating bananas does help the bananas last a bit longer as in up to 2 weeks even though the banana peeling may still start turning brown while in the refrigerator.
But when you do refrigerate bananas the refrigeration process does cause the bananas to lose most of their nutritional value.
It's best to just keep the bananas out on a shelf or on the counter and eat them before they go bad because the bananas will keep their nutritional value when out of the refrigerator.