No bears in that radius means the desert, and water in the desert means hiking along a stream.
An out and back on The Box or Death Hollow in Escalante comes to mind. It'd be nice in early summer, as hot weather would just let the kids go nuts playing in the creek.
If you could talk the park into giving you a permit, a 2-3 day thru walk of Lower Courthouse wash in Arches (from the park road S to the highway) would be great fun with kids. Lots of water, lots of rock formations to gawk at and play on.
There are many great options in the Needles, but you'd need to be water sherpa.