In that case, just keep enough cash to use for the day. There are ATMs EVERYWHERE in New York City, and use can also use your credit/debit cards almost EVERYWHERE. So really no need to have "emergency" cash... just make sure you keep one card separate from everything else.

A few months ago in Korea I busted the nail on my big toe on a white-water rafting trip. After paying the medical bills in cash I was running out of Korean won... this was a little town and none of the ATMs were on the international network, and it was the weekend so the banks were closed... so I just gritted my teeth, kept going around until I found a restaurant that would take my credit card, and waited until Monday morning to change out some travelers checks at a bank (also walked all around town trying to find a bank that would change them).

The alternative would have been using my last remaining won to buy a bus ticket to the next town over and hope that there was an international ATM there...

Moral of the story (although not useful for you): It's not just the third-world countries that don't have readily-available international ATMs!