How can I make my banana bread more moist?
Turn the oven down by about 25-50F. Quite often, ovens just blast the heck out of baked good, especially smaller apartment ovens. Larger, more expensive ovens tend to be better calibrated and will produce the proper 350F temperature usually required for banana bread. A cheap oven, which is still fine for banana bread, will overshoot the temperature and kill your banana bread. Turn the oven down a bit, and if necessary cook for longer. I have even turned my oven cooler AND cooked it for 15 minutes less time than the recipe required.
Check at 2/3 time. Don't let the bread run through the entire cooking time before checking it. Always check early. For some recipes, I take the banana bread out at 2/3 recommended time!
Don't over-beat the mixture. Despite being called banana "bread" it isn't a bread, it is a cake insomuch as it is made from a batter, not a long-kneaded dough. If you start mixing and it seems already too dry - STOP - and add a few tablespoons of milk to loosen the mixture up. If it's too dry in the initial stages, it will certainly be dry later, but a dry mixture also causes you to push harder, thereby causing the gluten to harden. 
Don't cut down on butter/fat. This is really what prevents it drying out. Use creme fraiche, more butter, a bit less flour...