If you file a joint federal return, Ohio requires you to file a joint state return. List yourself as resident and her as non-resident.
Ohio does a convoluted tax calculation for non-residents/part year
residents. It calculates tax on total income, then it calculates a non
resident/part year resident credit, which it subtracts from the tax it
calculated on the total income. The credit is calculated as your non-Ohio
income divided by Total adjusted Income multiplied by the total tax. TurboTax
(TT) does this by allocating your income as either Ohio or non-Ohio. W-2
income will be allocated by the state name abbreviation shown in box 15 of your W-2. TT
will ask you, item by item, in the state section, how much of your other income
is Ohio or
non-Ohio income. Make sure that her non-Ohio wages show MO (Other state postal abbreviation) in box 15 of your W-2 screen, with the MO amount
in box 16.
If your only income is military salary, and it was earned outside Ohio, you will owe no tax (the credit will be 100%).
