I wouldn't recommend taking the kid but not the wife. Doubly so if the kid's still on breastmilk. Some bases have BOQ housing available; some don't. Depending on your orders, you may or may not be eligible to stay in it. Saltier guys, please correct any bad gouge I'm about to give.
If you have TAD orders without authorized dependents, you should continue to receive BAH at your previous duty station, and also draw per-diem for the present duty station. You may use said per-diem for the BOQ, if you desire.
If you have TAD orders with authorized dependents, you'll get a single BAH for the present duty station. Base housing will likely not give you base housing with TAD orders, unless they're hurting for occupancy.
If you have PCS orders (these always include dependents, I believe), you'll get BAH for the present duty station. If you leave your family behind, that's your problem, not Uncle Sam's. The gov't housing you'll be eligible for assumes that your whole family is there, and will cost all your BAH.
When I say present duty station, I refer to the one that you're moving to as specified in the orders. Previous duty station refers to the one you left after receiving said orders.
In summary, when your orders authorize your family to move with you, they will pay for you to all live together at the new station. Don't expect special treatment for voluntary separation. On the flip side, if they force you to live separately, they'll pay you both separately to live where they expect you to live (at the previous and present duty stations). Whether or not your family continues to reside where they're "supposed to" can have a great effect on your bank account.