No, you do not need to file a state return for Kentucky (KY) if the only income from KY is wages. All income will be reported on your resident Ohio (OH) return.
Reciprocity:
A reciprocal agreement, also called reciprocity, is an agreement between two states that allows residents of one state to request exemption from tax withholding in the other (reciprocal) state. This can save you the trouble of having to file multiple state returns.
The local wages may not have any action for local tax withheld. Check with the KY city to see if there is a possible refund if local tax was withheld.