One corner of our backyard is designated for banana plants.

Don't just trim, the roots will stay alive and just grow back. You need to dig up or kill the roots.

Lucky you! Pot them up and sell them on and make a buck! or give them away..........That's the plant's nature.........

http://homeguides.sfgate.com/remove-pups-banana-tree-60208.html

I haven't been around banana plants since the 90's when we lived in Florida and had a couple. I use a good weed chemical to kill the wild grapes that are always trying to over run my yard here in Minnesota. Cut the offending ones down close to the ground and brush the cut end generously with the chemical. That makes it easier for it to get to the root than putting it on the leaves. It may take a couple of applications if it is stubborn. I use Round Up myself. Sed if this helps get rid of the extras.