Keeping bananas fresh for longer
Store them in a well ventilated place. Cool (15 degrees Celsius) and dark will probably do them good too. And buy green bananas. 

Your bananas are alive. Seriously, they continue to live after they have been picked. Breathing, cell metabolism, hormone production, etc. goes on. You cannot stop this process, you can slow it a bit. 

In many plants, fridge temperatures trigger their own growth slowing processes, because they are evolved to stop growing in winter months. But in tropical fruit, this doesn't work, because there are no cold months where they grow. Instead, it exposes their sensitive cells to an environment they are not meant for, and they change a lot. Bananas go black, tomatoes change their taste radically, and so on. 

Still, keeping them coolish and dark may do them good, because it still gives them less energy to spend on ripening.  

One of the mechanisms which controls ripening in fruit, including bananas, is ethylene gas. It is a positive feedback mechanism. Ripe fruit produces ethylene, and unripe fruit ripens faster when exposed to ethylene. People tend to stick green bananas in bags to make them ripe faster. If you can keep them well ventilated, the ripening will slow down. 

Still, don't expect any wonders. Bananas will ripe after being plucked, no matter if on the tree or not. And they will overripen in few days, even when held far from ethylene. Starting with green bananas gives you more headroom, but you still have a few days only.