How do I know if I owe state level taxes and need to make a payment with my extension in New York State?

Assuming that you have entered all your tax information and forms on your return, the numbers shown in TurboTax mean that you owe Federal taxes but are getting a state refund.

You'll need to pay the amount shown with your Federal extension. But you don't have to pay anything for state.

Thanks for the reply. I haven’t entered all my tax information and forms on my return yet … still working on my retirement contributions. As a matter of fact, I had to file for an extension today because I discovered that I made an excess contribution to my Roth IRA and will owe a penalty. In order to remove the penalty, I need to take action either withdrawing or recharacterizing the contributions from Roth  to traditional. Not sure how the excess contributions will affect my tax returns. Any thoughts? Or is it hard to say? 

Your excess Roth IRA contributions will not be taxed. However, if you have not withdrawn them by April 15, they are charged an excise tax of 6% of the excess contributions.