It was only stated that Worf wanted Troi to raise Alexander in the event of his death during the risky spinal cord transplant in "Ethics".
When Worf was off the ship, why would anyone need to watch Alexander? The kid's probably going to be in school for some part of the day, and during the rest of the day, he's as safe in his quarters as he would be in anyone else's quarters.
It's not the "where" I'm questioning; it's the "with whom". Technically Alexander is as safe as anyone else on the ship, but (for most of TNG anyway) he strikes me as too young to be completely unsupervised outside school hours. He's a smart kid but he's impulsive, and I'm sure that Federation rules regarding child welfare are pretty darned strict.