Shopgirl, at this time of year, if the parent is not growing new leaves, but instead putting out pups, I'd guess that the parent as well as the parent corm is getting ready to die. Banana's are bi-annuals after all. They're typically good for two years.
In the normal life cycle, after fruiting they decline and eventually die. I have a theory that banana left outside to over-winter, at times never fruit, but instead start dieing at the beginning of the season and start that process by shooting up pups.
When that happens to me, that I see that plant is no longer putting out new growth and I see a bunch of new pups coming up, I cut the plant to the ground. The reason I do this is that the psuedo-stem at this point will start to rot beggining at the top and will work it's way into the corm. By doing this I think I'm protecting the psuedo-stem from pre-mature rot and well as signaling to the plant that it's definately time to push pups out, because this "death life cycle" thing is really happening.
I'm far from an expert, perhaps more informed minds than my own will chime in, those are just some of the thoeries I've been kicking around lately.
gOOD lUCK!!
