You are on the right track.  You have to use a technique called allocating.  When you allocate, you designate the amount of income that you earned in each state.  That way, you pay the correct amount of tax to each.  

When you file the part-year returns for KY and OH, you will run into a screen that asks how much of the income was earned as a KY resident and then as an OH resident (on each return).  The amount reported on your W2 will pre-populate, and then you should be able to either override the amount or simply input into the corresponding box.

What you are seeing on the KY return is correct because you should receive a refund, since not all of the income was taxable in KY (only the portion of the year when you were a KY resident).  You should owe money to Ohio (since no OH state tax was withheld), but your KY refund might be higher than what you owe OH.  

"I worked the same job in ohio but my employer took state tax only out for kentucky. "

DanielV01 replied "The amount reported on your W2 will pre-populate, and then you should be able to either override the amount or simply input into the corresponding box"

If TurboTax(TT) is not giving you a place, in the interview, to change those amounts, you may have to use a workaround at the W-2 screen:

At the w-2 screen split the boxes 15-20 W-2 info into 2 lines. On the 1st line leave KY in box 15 but change  box 16 to the amount you earned while living in KY. Leave the KY withholding in box 17. On the 2nd line put OH in  box 15 and the OH portion of your wages amount in box 16; leave box 17 blank..