Why is part of the dividend exempt from state tax? Are you talking about a percentage of the dividend that is from U.S. government securities? If that's what you mean, here's how to enter it in TurboTax.
After you enter the information from the 1099-DIV, click Continue. The next screen asks "Do these uncommon situations apply?" Check the box that says "A portion of these dividends is U.S. Government interest" and click Continue. On the next screen you will be able to enter the U.S. Government interest.