When you owe taxes, be it to the city, state, or IRS, TurboTax does not, under any circumstances, with absolutely no exceptions, act as a middleman for that transaction. Payment of any taxes due is the taxpayers direct responsibility. If KY allows you to pay taxes due online, then you can do it that way. Otherwise, it is your responsibility to mail a check to the state.
From what I see, you can pay your KY state income tax online at https://revenue.ky.gov/Collections/Pages/E-file-Payment-Options.aspx