There was the great blizzard of 1899, trains chugged over the snowy Brooklyn Bridge a decade before that, and even more snow buried New York City streets and structures around the turn of the 19th century. Back then, there was no warning when a blizzard was on the way, so the city would have to scramble to deal with clean-up.
Snow removal was a daunting task at the time, without the more sophisticated equipment that we have today, removing snow meant shovels and carriages. They were still testing out methods of removal, and it wasn't always quick and efficient—in 1898, in a letter to the editor of the NY Times about "the snow removal problem," the reader noted: "Whether the tardiness in removing the snow from the city's streets is due to the lack of executive ability of the Street Cleaning Department or to the inefficiency of the force and the materials at that department's command, is an open question."
For the most part, you would see masses of snow shovelers out on the street (in this NY Times article from 1905, one is noted to have "fainted from exhaustion"), who would load snow into horse-drawn carriages. From there, the snow would be dumped in the East River (which is not allowed anymore).
By 1902, the Times reported that the Department of Street Cleaning had 2,500 "of its own men and 1,500 hired helpers, [who] undertook the problem of removing from the city streets of snow and ice which had accumulated." Too bad they didn't have flamethrowers, but a decade or so later they did get some motorized plows.