Bananas should be left to ripen at room temperature, but, according to several sources, including Dole Food Company, ripe bananas can be refrigerated. While this will dramatically slow the ripening process, be aware that the peels will turn black.
Someone of a certain age saw a banana in my refrigerator and asked what I was doing.
"Haven't you heard the Chiquita banana song?" he asked. "Never put bananas in the refrigerator!"
A little Googling turned up this gem of post-World War II marketing: