I'm active duty and my wife is enlisting. Will the army expect me to move into the barracks when she goes to basic training?

No because you are legally married and collecting BAH. She may change to not be a "dependent" but then you would receive BAH at the without dependent rate.

She will remain your dependent until she graduates AIT. She will not receive bah during her BCT or AIT. Only after she graduates and PCS’s to her new duty station (whether that be there at Bragg with you or some where else). After that, assuming you two don’t have any children, you will both receive BAH without dependent rate. Make sure she enrolls in the MACP program before she finished AIT that way you two have a better chance at staying together. Good luck!

They shouldn't. Even if she's currently in training, you're still married. If they do try to move you, they could do it everytime you or her goes to training.