No, the Ohio local taxes are not a credit on the KY state return. And, because Ohio and KY are reciprocal states, they have an agreement that when you work in one and live in the other, you are only taxed where you live. If Ohio state taxes were withheld, you will file an Ohio nonresident reciprocal return to get these taxes returned to you. However, Kentucky is taxing all of your income. Because of the reciprocal agreement, you will not need the credit for tax paid in Ohio because you will get all of that back. Local taxes are not a part of the reciprocal agreement, and KY localities only tax you when you work in that locality, so there is no credit to claim for your Ohio local taxes.
You may fill out a form with your employer so that they withhold KY and not OH tax. Please see the following FAQ for more information on this: https://ttlc.intuit.com/replies/4776822
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