The polite refusal answer in the New York Times crossword puzzle is typically given as "NO SIR." It is a two-word phrase used as a courteous way to say no, especially when addressing someone with respect.
For a five-letter polite refusal answer, "NOSIR" is common, which is a contracted form of "No sir" used in polite refusals.
Thus, a polite refusal fitting the New York Times crossword style would be "NO SIR" or "NOSIR" depending on the clue length and formatting. Both represent a respectful way of declining or saying no politely.