State tax filing

With SC in box 15 the income and withholding was reported to SC.  There will be no correction; the W2 is reporting what the employer already did.  On your tax return, however, you get to pay the right taxes to the right states.  

NY, your non-resident state, only taxes NY source income.  SC is your resident state and taxes income no matter where it is earned.  However, you can claim a credit on your resident SC return for taxes paid to another state to mitigate the double taxation.  

As @PaulaM stated you will need to file a non-resident NY return to report the income earned in NY and since there was no withholding you will likely owe a tax there.  Once you complete the NY non-resident return take note of the income taxed and the NY tax liability.  Those are the numbers TurboTax will use to help you claim a credit on your resident SC return. 

The SC withholding and the credit for taxes paid to another state will reduce any SC tax you may owe and possibly yield a refund depending on your other income and withholding amounts. 

Here are the steps to generate the NY non-resident return.

1. In the "My Info" section
2. Scroll down to "Other State Income"
3. Answer "Yes" you earned income in another state
4. Select "NY"
5. Continue
6. After entering your income and deductions in the federal interview
7. Start the NY interview
8. Complete/Verify the residency questions
9. Next will be the income questions where you will allocate the W2 amount that belongs to NY
10. Any other non-NY income sources be sure to allocate zero "0" to NY
11. After completing the NY interview then you can work the SC resident state interview
12. In the credits section is where you will find the credit for taxes paid to another state