You only include the amount that was used for qualified educational expenses. The whole amount from BAH may not be included because it is not required to pay for educational expenses.
Example from IRS Publication 970 states:
You have returned to college and are receiving two education benefits under the latest GI Bill: (1) a $1,534 monthly basic housing allowance (BAH) that is directly deposited to your checking account, and (2) $3,840 paid directly to your college for tuition. Neither of these benefits is taxable and you do not report them on your tax return. You also want to claim an American opportunity credit on your return. You paid $5,000 in qualified education expenses (explained in detail in chapter 2 ). To figure the amount of credit, you must first subtract the $3,840 from your qualified education expenses because this payment under the GI Bill was required to be used for education expenses. You do not subtract any amount of the BAH because it was paid to you and its use was not restricted."