New Orleans was indeed a major port - big enough to ship directly to and from. With the introduction of packet shipping lines especially, much of the cargo from other Southern ports like Pensacola, Mobile, Charleston, Savannah, etc etc was carried up to New York, unloaded, warehoused, consolidated, repacked onto larger ocean going vessels and then sent off across the Atlantic.
What New York offered in addition to a large shipbuilding industry were a lot of export services like banking and insurance and the "Factors" (today we'd call them middlemen) who arranged a lot of the business side of exporting. The Northeast but specifically New York actually made more money from these business services than the Southern Planters made from producing the cash crops. The Imports flowed through the same channels in reverse. Plenty of those coastwise trade vessels carried cash crops north and manufactured goods South. The loss of this business in addition to the loss of tariff revenue and the loss of the Southern states as a captive market for manufactured goods would have been an utter disaster for the Northeast. This is why their editorials quickly turned extremely bellicose. Before the business "movers and shakers" got involved, there were plenty of Northern editorials saying essentially that they should let the Southern states go in peace.
If New Yorkers made more money from shipping than cotton growers did, a reason might be that they were also shipping many other goods to and from Europe, not just cotton. Another reason why New York prospered might be that the cotton growers could never be sure they'd have money when they needed it, so they were a market for loans and credit from New York banks, the economy of the cotton states being so dependent on cotton that Southern banks may not have had the money when the planters needed it.
New York and the Northeast did fine economically after the slave states seceded. Among those who were most inclined to let the secessionists go in peace were those who did business with the South. They assumed that, given peace, business relations would go on much as they had. They appear to have been more worried about the Southern states renouncing their debts than about tariffs or the South yanking away their business, and they weren't at all keen on war. The mayor of New York even wanted the city to secede from the United States and keep up the cotton trade with the Confederacy. The attack on Fort Sumter changed that. Source