Conjecture: Europeans do not freak out about a picked pocket or car break in and shrug it off. Americans seem to have zero tolerance about petty crime and obsess for "safety," and ask endlessly the question "is _____safe"?
That is not a fair statement at all.
First - I doubt European victims of petty crime just shrug it off.
Second - For them, it ruins their day and they are out whatever cash they had and the inconvenience of replacing lost cards and IDs. But for a foreign tourist, it has the potential of ruining the trip, or at least messing with tightly scheduled plans that people have spent a lot of money to come see. People might lose their photos, their passport, and their way of paying for things for at least a couple of days.
Third - Most Americans don't understand the nature of crime in Europe. When we hear about robberies in the US, they tend to be muggings and threats of violence to turn over valuables. Whereas in most of Western Europe, robberies are stealth-based like pickpocketing, stealing purses left on chairs and breaking into unattended cars.
And to add to the earlier answers, while I have no doubt that NYC has some pickpockets, they are probably amateurs compared to those in many European cities. The idea of being alert or sticking your hand in your pocket won't work because they prey on the times you are not alert and don't have your hand in your pocket. And they often work in teams to create distractions and keep you from being alert. The chances are still low of being a victim, and it varies significantly by city, but it still happens enough to keep the lines at the embassies busy for replacing passports and such.