I did a little looking into this, and from what I have found, it's sort of true. Christmas was a much more public affair than it is now, resembling Carnivale in the fact that it was largely celebrated in the street with lots of drinking and shouting. In fact, during the 1820s there was a cultural battle of sorts over the nature of Christmas. It was changing from a paternal holiday, in which workers were given/demanded food and drink from their bosses to a more private holiday in which families gave presents to their children. Changing the nature of the holiday from a focus on class to a focus on age. This transformation was not easy, however, and throughout the first half of the nineteenth century there was a lot of crime and trouble around Christmas.

In 1828 specifically there was a particularly notable Christmas riot in which a ""callithumpian parade" complete with "drums, tin kettles, rattles, horns, whistles, and a variety of other instruments" ranged around New York, beginning in the working class neighborhood of the Bowery, continuing to Broadway, then to a black neighborhood where they "demolished all the windows, broke the door and seat," and beat African American congregants at the local church with sticks and ropes. The group continued to range around the city breaking things, including the windows of wealthy citizens' houses. As they made their way back to Broadway they found a group of hired watchmen were waiting for them. However, these watchmen were not exactly professional policemen, being more of an ad hoc organization, and anyway the mob actually faced them down and continued along its destructive way. *

The first professional New York Police Department that I can find was made in 1845, led by a man named George Matsell, a New Yorker who had worked his way up through the Tammany ranks. Even as late as 1845, there had been considerable resistance from the citizens regarding the creation of a police force and it was not a popular decision.

Sources: Nissenbaum, The Battle for Christmas, Stansell, City of Women, Lankevitch, American Metropolis