Will a state tax credit be automatically applied to the another states owed taxes?

That's backwards.  New York will not give you credit for taxes paid to Connecticut.  Connecticut will give you a credit for the taxes you pay to New York, at least up to the point that Connecticut taxes the same income.  

For this to calculate correctly, you will prepare the New York nonresident return first.  This should transfer the credit automatically.  But pay attention to the actual amount of tax to New York, which will be any withholdings plus any additional tax paid or minus refund you will receive.  That way if it doesn't pull over, you can enter the amounts in the Taxes paid to another state screen.