NO Prozac, certainly not in the doses you mention, and no crates. You need to build her confidence, not destroy it. It seems to me the poor little dog is grossly over-stressed.
Work with her, as Janja suggests. Quality time with her, longer walks, some kind of training that is special to her - obedience, even agility if you have a child old enough to do it with her. Fun for them both.
Children need to realise that when a Basenji needs space and time, they have to respect that and leave the poor animal alone. Unwanted attention at the wrong time WILL be rewarded with a growl. At first (often) a low murmur, just a warning. But for Pete's sake don't let it degenerate into something more serious.
My constant moan is that very few breeders crate train before the pups leave the nest. At 3, it will be more difficult and so much better to set aside 'HER' room where she is undisturbed, lots of toys, maybe, and a favorite bone to chew on. (Beef bones, raw, will last for ages and clean her teeth the while). And nothing which you cherish.
You are not going to lick this problem over night. It will take time, disciplining the children, lots of patience and a whole bundle of love.
Reduce the stress, eliminate the medication, lock the crate away and start to spend time building the wee girl's confidence again.