"Had pass-through state income tax from K-1" - How do I know if I had this for our S-Corp and where do I find the amount passed through?

If you had state taxes withheld on your K-1 from an S-Corp, it would appear on Box 12 with a code of S (other deductions).  After entering the amount in box 12, you will be asked to enter details.  Select the state name from the list and enter the amount of state income tax withheld.  See screenshot below.