Russell Shorto's book The Island At the Center of the World takes a couple of anecdotal statistics from Narratives of New Netherland to give early population estimates. A writer of a November 1626 account of the colony estimated that there were about "two hundred souls" who made up the population -- this was shortly after the permanent European settlement had been established.

It was slow-going at first, under the early leadership of Wouter van Twiller, Willem Kieft, and the other initial Directors-General of the colony. By 1646, a contemporary account estimated the Manhattan colony was populated by about "four or five hundred men".

More information is found on pages 48-49 of the book Colony of New Netherland: A Dutch Settlement in Seventeenth Century America by Jaap Jacobs. Jacobs (as have others) notes that most of the colony's population growth during the Dutch period happened in the last fifteen years, under the leadership of Director-General Peter Stuyvesant, often credited with resuscitating a colony that still risked failure, twenty-odd years after being established. He notes that one estimate gives a population in 1664 (right before the English takeover) of 6,030 colonists. But Jacobs argues "in my opinion and that of other scholars" this estimate "is far too low", and was probably closer to 7,000-8,000 colonists, and as high as 9,000. In New Amsterdam (Manhattan) itself, the population was probably around 2,500.

BONUS:

Under English (and, later, British) rule, the population continued to grow substantially. The two books you may want to consult are Population History of New York City by Ira Rosenwaike, and The Population of the British Colonies in America before 1776 by Robert V, Wells. According to Rosenwalke, by 1698, thirty-four years after the English takeover, the population of New York City (which only included Manhattan at the time) had grown from about 2,500 to just shy of 5,000. By 1749, the population of the city had grown to over 13,000. In 1771, just a few years before the Revolution, the population was approaching 22,000.

From 1790 onward, the population numbers become more accurate, as the U.S. has conducted a census every ten years since then. On that first census, New York City's population stood at 49,401 -- more than doubling since the Revolution. This was due, in part, to New York's role in the war. It was occupied by the British for the duration of the war, and once they left, Patriots flooded into the city, to reclaim property or to make a new go of it.

From then on, the population of the city exploded, generally tripling about every thirty years to the end of the 1800s. I have written in this sub about what factors led to New York overtaking Boston and Philadelphia in the late 1700s/early 1800s as the commercial center of the United States.

Also note that there was a big bump in the 1900 census, because, in 1898, New York City (Manhattan) merged with Brooklyn, most of Queens, Staten Island, and the parts of the Bronx not yet consolidated into the city.

Suffice it to say, we have a pretty accurate picture of how many people were living in New York City from 1790 on. The earlier, pre-1790 numbers are calculated using partial censuses, tax lists, church parish records (calculating based on number of baptisms vs. number of burials), and other available records from the era.