Are you using tap water to water the plant? If so, I would fill up a container with the water and let it set for 24-48 hours. This will allow some of the chlorine and other chemicals in the water to dissipate or "gas-off." Then use the water on the plant. I'm new to bananas, but my other indoor tropicals used to get the black edges on the leaves until I let the water sit for a while.

I am in zone 6 - Kansas - and I lost a couple of smaller pups earlier this summer that turned black and then just withered away. I believe it's caused by soil that does not drain well and overwatering. The overwatering was not entirely my fault as we had alot of rain in May and June. The plant has since thrown a couple of new pups and with the drier ground these pups are doing fine. Good luck.

Thanks for the tips. I am considering repotting it in a soil/sand mix instead of the soil/peat mix that it's in right now. I will also try the rain water trick. Any other ideas would be greatly appreciated !