What's reported in box 15 doesn't necessarily mean anything, but if there's a corresponding amount of tax withheld in box 17; then you'll definitely need to file for the state.
However, if you lived and worked in MN, you don't need to file a NY return just because your employer is located there (unless state tax was withheld to get it back). If your box 15 has MN, but there's no corresponding amount in box 17 leave box 15, employer ID and box 16 blank. If your box 17 has an amount, fill in the box 15 -17 exactly as on your W-2. In either case, you'll need to file a MN tax return.
Let me know if you have any follow up questions.
