Hendrik Van Loon, in the "If: or, history rewritten" collection had a bit called "If the Dutch had Kept New Amsterdam": the Dutch either cede northern and western NY to the Brits or fail to occupy them, creating a connection between the southern and northern colonies, but New Amsterdam survives as an independent enclave.
It becomes a major center for smuggling weapons during the revolutionary war (which, as in our TL, leads to the independence of all the colonies south of Canada), and, IIRC, it becomes independent itself during the Napoleonic occupation of the netherlands.
As time goes on, the population becomes increasingly anglicized due to extense inflow of US citizens, and in 1841 it becomes part of the US. It however does maintain some special legal privileges of it's own - which becomes manifest during Prohibition, during which it uses a resulting legal loophole to become the only non-dry state in the Union...
Bruce