Avoid deserted blocks in unfamiliar neighborhoods. A brisk, purposeful pace helps deter trouble wherever you go. New York City is a safe city, but it's still a city, so keep jewelry out of sight on the street; better yet, leave valuables at home. Don't wear gold chains or large jewelry, even if it's fake.
When in bars or restaurants, never hang your purse or bag on the back of a chair or put it underneath the table.
Never leave any bags unattended, and expect to have you and your possessions inspected thoroughly in such places as airports, sports stadiums, museums, and city buildings. Police officers stationed by subway token booths also reserve the right to check your bags before you pass through the turnstile to enter the platform.
Politely ignore panhandlers on the streets and subways, people who offer to hail you a cab (they often appear at Penn Station, the Port Authority, or the airport), and limousine and gypsy cab drivers who (illegally) offer rides priced according to how desperate you look.
Knockoff wristwatches will keep excellent time until you're about an hour away from the vendor, so don't bother with them; ditto for pirated DVDs. Trust us, they work poorly and, not for nothing, their sale is highly illegal.
If you wander into a cold snap, do as the locals do and buy an inexpensive hat or scarf from a sidewalk vendor. Similarly, if it rains, scan the mouths of subway stairwells for umbrella salesmen who materialize so quickly you'll think the raindrops hydrated them into existence.