The rise of cotton affected New York by helping the city’s economy tremendously. The article in the link below says that New York was, in 1860, “the capital of the South” because of the fact that it was the center of the cotton trade.
New York was the center of the cotton trade in part because it was a transshipment point for cotton. There was a “cotton triangle” where cotton was shipped from the South to New York on coastal ships and was then transferred to larger ships to be taken across the ocean to Europe. This helped to make New York’s economy boom.
This was not, by any means, the end of cotton’s importance for New York. Instead, New York made money off of cotton in many other ways. It provided insurance coverage for the cotton cargoes that were shipped. It provided financial services for the shippers and others connected to the cotton industry. It also produced many goods that were shipped to the South to be sold to the planters there since the South had very little industry of its own.
In short, New York was central to the South’s economy. Since the South’s economy was based on cotton, this meant that much of New York’s rise to prominence was based on the money it made from its various connections to the cotton trade.