I got a case of bananas for the first time a couple weeks ago. They were what the guy said "perfectly ripe", which mean they were green, but not too green. :) I found out that what I call ripe they call "over-ripe". I like getting the terminology straight here.
The bananas were organic, at 99c/lb + 10% discount. It's New England, what can I say, everything is expensive.
They came in a cardbox, and were wrapped in a big plastic bag. And here my good tips commence...
Bananas do not like to be stored in a non-breathable space or container. In a plastic bag, they became mushy and started falling off the bunch before becoming ripe. Also, the bag became all wet, even though it had some wholes in it. I suspect if I left them much longer in that situation that they would become rotten and/or moldy before ripening.
I went shopping without my reusable bags, so I got a bunch of paper shopping bags so I could use them to store my bananas. Paper bags are breathable. I cut out the handles of those bags so the bag would remain rolled in easily, and placed enough bananas on each so as to not to pile any banana on top of other bananas. This helps because as they ripen they don't squish and "hurt" the bananas at the bottom, which could start developing a rotten patch and attract fruit flies (if they were able to get in, maybe strongly motivated by rotten fruit).
I stuck the bags under my buffet furniture, since it had enough space on the bottom to keep the bags rolled in and it was a convenient place for me. As a bonus, no fruit fly was able to get into the bags.
That's it. I had to throw out some that had gone too mushy because I took long to intervene. With my next case I believe I will have it down perfectly. The bananas ripen really fast in the paper bags.
Another tip: the ripened the banana, the faster they will continue ripening.