First, I want to say you are to be commended for your concern for your neighbors; not everyone is so considerate.
I'd suggest that what you need is a backboard that doesn't vibrate and hence won't radiate noise. A sheet of lead would be ideal, but might be hard to hold up, so I'd suggest this as an alternative: you can make the backboard from a sheet of "dB-Ply" plywood from Greenwood Forest Products (Portland, Or.) It is a plywood that incorporates a viscous damping layer.
Others have suggested foam or other materials sprayed to the backside of the backboard and while these will slightly reduce the noise, they are not as effective as you might wish.
What you want is a means of applying a constrained-layer damper to the backside of the backboard: one way of doing this would be to apply a layer of 3M trim cement or equivalent to the backside, and use it to mount a layer of sheetmetal or plywood. Then vibrations in the backboard are dissipated in trying to shear the adhesive.
Then you'll only have to deal with the kids' shouts when they score the winning basket...