I live in Tennessee but work for company based in Cincinnati, Ohio. Do I have to pay Ohio and Cincinnati income tax? I work there a few days a month.

[TomD8]

Yes, you have to pay both Ohio and Cincinnati income tax.

Ohio non-residents must file an Ohio tax return if they have any Ohio-source income.  Income is "sourced" based on where the work is actually performed.

Cincinnati levies its income tax on anyone who lives in or works in the city.

[glenncarter1962]

Is it pro-rated, since I only work there a few days a month?

[TomD8]

Here's a link to Cincinnati's income tax webpage:  <a rel="nofollow" target="_blank" href="https://www.cincinnati-oh.gov/finance/income-taxes/">https://www.cincinnati-oh.gov/finance/income-ta...>

[Hal_Al]

Yes, it's prorated. You will have to file a non resident OH return and pay tax only on the income earned when you physically worked there.
The same rule applies to Cincinnati city tax. But, if  your employer is withholding Cincinnati tax, you do not have to file a city return, unless you want to get a refund if he withheld for days you didn't work there.