I had a refund on my state return last year that was unpaid. Where on my 2020 Kentucky state return do I put the unpaid refund from 2019 ?

What do you mean it was unpaid?  Did you ever receive the refund?

If not, you may want to contact Kentucky to ensure your return was received and processed.  They would have issued you a 1099-G for any refunds issued from your state tax return last year.

Track your Kentucky refund

Unless you elected to have the refund applied to your 2020 income tax return, you would not need to enter anything on your state income tax return in 2020.

If you had a refund from 2019 that you applied to your 2020 tax return, you can enter this information in the federal section of the interview. 

1. Select Deduction & Credits 
2. Scroll down to All Tax Breaks 
3. Select Estimates and Other Taxes Paid 
4. Select Income Taxes Paid 
5. Select Yes and continue to this screen.  Select 2019 refund applied to 2020 state taxes