If I am married and my wife does not move with me, will I be able to receive BAH for her and still live in the barracks?

From what I read last year, they stopped geographical bachelor's in the barracks.
https://www.thebalancecareers.com/army-announces-end-to-geographic-bachelors-3353934

According the Army Regulation 420-1, Army Facilities Management page 68, “Soldiers entitled to BAH at the “with dependent” rate, who are voluntarily separated from their Family members, are not authorized assignment to Un-accompanied Personnel Housing (Permanent Party) in the continental United States, Hawaii, and Alaska. This affects all permanent party categories and all ranks except for chaplains as set forth in paragraph 3–18e. Garrison com-manders retain the authority to identify and assign key and essential person-nel to UPH.”

This being said Geographical bachelors are required to live off the installation at the BAH with dependent rate for the location that they are stationed at.

So you will not be able to live in the barracks according to this change. She doesn't get BAH - you get the BAH. What you do with your BAH is up to you.

It all depends on whether or not your unit has spare rooms. No one will know this unless you know the exact unit you are going to. If you are receiving BAH, they do not have to give you a barracks room. The priority obviously going to the single Soldiers. Have a plan for the worst case scenario.

Like anything involving finance, your question has multiple answers.
When you arrive at your duty station you will receive BAH for your duty station location, not your spouse's. Since she lives in San Diego, it will probably be up to $1000 less. You will not be authorized to live in the barracks at all.
If you go overseas you may leave her and receive BAH for her location while you're gone.
Here's an article explaining the geographical bachelor thing.
https://www.thebalancecareers.com/army-announces-end-to-geographic-bachelors-3353934