The Dutch legacy of religious tolerance, multiculturalism, representative democracy, free trade, etc. are much vaunted in some discussions of the development of New York City (cough Russell Shorto cough). And yet a major impetus for handing it over was the press of English settlement, which presumably only increased after the transfer. So how well did these ideas really survive beyond the crafting of the city charter? Did these political traditions leave marks beyond New Amsterdam?
Great question. It's complicated.
Even though New Amsterdam was surrounded by English ships and was obviously going to be defeated, the Dutch governor Peter Stuyvesant did not surrender the colony of New Netherland until the English agreed to his articles of capitulation.
These articles stipulated various rights for the Dutch colonists under the rule of their English captors. To answer your question in short, you can see that articles 8 and 10 were reproduced almost exactly in the Bill of Rights a century later.
To give the long answer, the Articles of Capitulation granted rights to the Dutch citizens of New York that set them apart from the other colonial subjects in North America. That is, if these rights were acknowledged. Just after the surrender, it seems that the King of England told the new governor of New York (Andros) to ignore these rights, and for instance forbade commerce between the colony and the Netherlands (which was a condition promised by the articles). The governor soon found that it was impossible to rule the colony without granting some level of autonomy to the Dutch and letting them continue having such rights. Though the government of New York was officially supposed to be absorbed into that of New England, reality proved that its own animal, and the Dutch continued to have distinct privileges.
In 1673 a Dutch fleet invaded and conquered the colony, which was still mostly populated by ethnically Dutch people, and abolished English law in favor of Dutch law. Though it lasted only 15 months, the Dutch community welcomed this event, and it reinforced the division between them and the English.
About 20 years later there was an event called Leisler's Rebellion, which resulted in a faction with strong allegiances to the Dutch, that was opposed to its English rulers, once again gaining control of the colony. It remained a de facto independent state for three years. Though by now the original division between the English and the Dutch was becoming blurry, it was another episode that illustrated that the residents of New York had their own unique ideas about government and were not bound to obey the English rulers of the colony.
These events established a strong precedent in the colony for the idea that its residents could justifiably rebel against their English rulers, which certainly proved to be an important idea during the revolutionary war. This distinction between the populace and their rulers is perhaps the most significant legacy of the Dutch colonists. The idea that the populace had certain inalienable rights can likewise be traced to the Dutch, and as was said before, the claims to some specific rights seem to have survived from the end of the Dutch period to the American Revolution.
The influence of this Dutch legacy on the Revolution is not often discussed, but is significant. It seems that not only was the model of a Republican government partially based on that of the Dutch Republic, but even that the Declaration of Independence was written in imitation of a Dutch document, and was printed on Dutch paper. Not only that, it is notable that New York was was the first capital of the newly independent states, and that the Dutch Republic was a close ally of the United States during its initial years.
Brodhead, Documents relative to the Colonial History of New York, Vol. II-III. (1858) McCormick, Leisler's Rebellion (1989) Rosenblatt & Rosenblatt, Opening Statements (2013)