You may be entitled to a credit for taxes paid to Ohio on your Kentucky tax return, that is why TurboTax is asking for that information. You can find the amount by looking on your W-2 form that was issued by the company you worked for in Ohio. 

On line 15 you will see an abbreviation for the state, and line 17 and 19 will be the amount of taxes you paid there. If there is no Ohio tax listed, then you should enter "0" for the amount of taxes paid to other states.

You can NOT get a credit for taxes paid to Ohio on your Kentucky tax return. If OH taxes were mistakenly withheld you have to file an OH return for a full refund.

But, if no OH tax was withheld, you should not be messing with OH at all, in TurboTax.

In the personal info section, when it asks if you made money in another state, answer NO, per the reciprocal state instructions, on that page. This will prevent TurboTax from thinking you need to file an OH return.

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Back Ground

 Ohio has a reciprocal agreement, on wages, with all neighboring states. No Ohio state taxes are withheld or due and you do not need to file an OH return. But your home state will tax you on your Ohio income.

If Ohio taxes are mistakenly withheld, you have to file an Ohio return to get a refund. You should ask your employer to stop withholding so you don't have to file an Ohio return every year.

However the reciprocity agreement does not apply to local city taxes. You are not required to file a local city return, but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount. If you are subject to a KY county or city tax, you should check to see if the KY city will give you a credit for the OH city tax withheld. KY does allow an itemized deduction (Schedule A) for local tax paid.