It's strongly hinted at that the more adventurous crews reach Central America and the Caribbean, possibly even South America.
As David's uncle Axel relates:
And yet it can't always have been like that because there was one ship whose captain was foolhardy enough to sail close inshore. Her crew were able to make out great stone ruins. They were all agreed that they were far too regular to be natural, and they thought they might be the remains of one of the Old People's cities. But nobody knows any more about them. Most of the men in that ship wasted away and died, and the rest were never the same afterwards, so no other ship has risked keeping close in.
For hundreds of miles the coast goes on being Badlands with stretches of the dead, black lands; so far, in fact, that the first ships down there gave up and turned about because they thought they would never reach any place where they could water and provision. They came back saying that they thought it must go on like that to the ends of the earth.
In the middle of all the fuss, however, a ship called the Venture which had long been given up for lost, came sailing home to Rigo. She was battered and undermanned, her canvas was patched, her mizzen jury-rigged, and her condition foul, but she triumphantly claimed the honour of being the first to reach the lands beyond the Black Coasts. She brought back a number of objects including gold and silver and copper ornaments, and a cargo of spices to prove it. The evidence had to be accepted, but there was a lot of trouble over the spices, for there was no means of telling whether they were deviational, or the product of a pure strain. Strict churchgoers refused to touch them for fear they might be tainted; other people preferred to believe that they were the kind of spices referred to in the Bible. Whatever they were, they are profitable enough now for ships to sail south in search of them.
You'll find islands where the people are all thickset, and others where they're thin; there are even said to be some islands where both the men and women would be passed as true images if it weren't that some strange deviation has turned them all completely black -- though even that's easier to believe than the one about a race of Deviations that has dwindled to two feet high, grown fur and a tail, and taken to living in trees.
The Chrysalids, chapter 6.
The islands where people are "all completely black" might be some of the Caribbean islands. How uncle Axel describes them, they seem to be at the furthest edges of where the Labradorian ships have sailed. The spices these ships brought back can easily have come from Caribbean or Central American countries.
Ships that come (too) close to the coast, see "great stone ruins" the ruins of cities, thought to be "the remains of one of the Old People's cities". So while there's no specific description of New York City, it's safe to assume it lays in ruins.
The "race of Deviations that has dwindled to two feet high, grown fur and a tail, and taken to living in trees" sounds like an accurate description of new world monkeys, found in Central and South America.