To keep a bunch of bananas fresh for longer, wrap the stems in some plastic wrap. Re-cover the bananas with the wrap after removing one.
This method prevents ethylene gas, produced naturally in the ripening process, from reaching other parts of the fruit and prematurely ripening it. This technique is hit or miss, as the coverage from the plastic wrap is unlikely to fully prevent contact with the ethylene gas. It's certainly better than nothing, though.
This explains a few common tricks about using bananas to ripen other fruits like avocados. Or quick-ripening bananas by storing them all in a bag together. Ethylene is actually used in the banana production facilities to induce ripening at just the right time to ensure that you buy a bunch of yellow (or greenish yellow) from your local grocer.
(The next step is my preferred method, and the one that the science appears to back up with the most evidence.)