Do I have to file a New York tax return if I only made $2020 in New York income, and why when I file does it say I owe income tax when $2020 is less than the deductible?

You can delete New York from your State Retuns (screenshot).

Here's detailed info about filing New York taxes if you are a Resident, Non-Resident or Part-Year Resident:

https://www.tax.ny.gov/pit/file/do_i_need_to_file.htm

Edited 4/8/19 1:31 pm

I've looked at that, but it doesn't answer my question. It says I need to file if my New York income exceeds my New York Standard Deduction, which is $3100- so it doesn't. So theoretically I shouldn't owe any taxes or need to file- but when I put the numbers in TurboTax, it says I owe $37 in New York taxes. This doesn't make any sense and I'm wondering why the website is saying I owe tax when my income is less than the standard deductible so there's no reason I should owe tax.

Since you know you are not required to file a NY return, you can delete it from your State returns. See screenshot in Edited answer.