Eaglelox, generally bananas are picked very unripe. In summer time, they ripen very fast. However, in winter the ripening process is much slower, so if you buy then green now, they will take quite some time to ripen if you keep them in a cool place. If you do anything more drastic such as putting them in the fridge, they may stay unripe but you run the risk of having them become darker in colour and eventually failing to ripen when you want them to do so.
Alternatively, ripe bananas can be kept in the fridge which will preserve them longer at the expense of changing their appearance and possibly texture. However, they will most likely be okay to eat regardless.
The proof of the pudding is in the eating!