The best way to keep bananas for a longer time is to make sure that they
are in a cool, dry place (not on top of the fridge) and to *hang* them,
rather than to lay them on a flat surface or in a bowl. There are actually
little "banana stands" built for this purpose that you can place on the
countertop and then hang the bananas over the hook on the top. I've always
had luck simply taking a piece of twine about 12" long, typing a knot in it
to form a big loop, then stringing the bananas over the loop and hanging the
loop from a cup hook under the cabinet so that the bananas are suspending
over the counter top.
The reason that bananas will keep longer if they are hanging is that when
they hang, the gases that are given off during the ripening process can
escape away from the banana; when these gases are trapped in the vicinity of
the banana, like when the bananas are sitting in a bowl, they speed the
ripening process. That's why you can put a banana in a paper bag with other
fruit (especially peaches and avocados) to hasten the ripening process of
the other fruit.