It depends on the amount.  See "Who Must File" on page 7 of this reference: https://www.tax.ny.gov/pdf/current_forms/it/it203i.pdf

OK.  I read it. I'm still confused. My net NYS income is zero ($1698 income in Box 7 of a 1099-MISC to reimburse $1698 in travel expenses).  TT is telling me I owe $84 in NYS taxes because my NYS AGI (all from non-NYS sources) is greater than the NYS standard deduction.  So with ***ZERO*** NYS net income, TT is telling me I owe $84 in NYS taxes???  This doesn't make any sense.  

Was that meeting held in New York?  Otherwise it would not be NY-source income and not taxable by NY.

If the income was NY-source, you're correct: you should still owe no NY tax (even though you'd have to file an NY return).  TT should have generated a Schedule C or C-EZ when you entered the 1099-MISC with Box 7 income.  On the Schedule C or C-EZ you would deduct the entire 1099-MISC amount as a travel expense, resulting in zero net income.  Note that NY uses your total federal income to calculate your NY tax rate, but it applies that rate only to the NY-source income.