I have been a poll worker, though not in NY. In my state the poll workers could have made it a spoiled ballot (lots more work for the poll workers), offered a paper ballot as they did, or asked him to wait. It would also depend on how many machines they had, and their set-up, what kind of machine they had, and whether they could simply change cards in the machine.
The audacity of which I spoke was their assumption that the voter was a democrat when the voter first came in. Not only did they assume that voter was a dem, but also the next one as well. I've voted my many different places since turning 18 (and in three different states), and never once has anyone not asked me what party I was affiliated with when I came in.