A gadget based on the toilet tank float principle could automatically cut off water input when the level reached a certain height.

Or better, the level mechanism used in clothes washers, possibly in conjunction with low voltage solenoid valves.

Any connection between plumbing and electricity invites disaster. Better a wet floor than electrocuted kids.

Sounds like a great idea. They make little battery powered flood alarms to put in basements and such which could be easily rigged to do what you want. They're inexpensive and about the size of a Walkman. You can get them at most hardware stores. Being battery powered, there is no electrocution hazard. You merely need to attach it to something that you can hang on the side or stand up in the tub. Most tubs are made of ceramic or plastic over steel, so you could possibly stick it to the side with a strong magnet. A suction cup might also work, but I personally wouldn't trust that as much.

Good suggestion, Dwane. I suggest you (or someone) go to the site of the mfgr. of the basement flood alarm gadget and suggest they include an attachment gadget, or mention in their user manual how the user can make one from Velcro, to allow it to be used as a bathtub monitor as well. This would increase their sales.