Do you need to fill out the Nonresident Business Allocation Schedule to allocate New York State income?

If you lived in NJ but worked in NY, you have to file a non-resident NY return. 

New York collects taxes on income you earned while working there and, as NJ resident you will be taxed on all your income. To avoid double taxation, you will claim credit for taxes paid to NY on your NJ resident

I'm in a similar situation, my employer in based in NY and I live in MA.  Your comment makes sense how/where to pay taxes, however you didn't directly answer the question to whether or not to fill out the "Nonresident Business Allocation Schedule" form IT-203A.   

Do we need to fill out that form? The language on that form makes it seem like that form is for the employer not employee. 

Thanks

Yes, you would complete the IR-203-A. Here is an article about how to get TurboTax to guide you when you have multi-states.

@TimInMass