You're thinking of New York & San Francisco, both of which had extensive pirate/privateer caves/tunnels carved into the bluffs overlooking the water. There is a "street" called Pirate's Alley in New Orleans, but that's about it. Pirates in New Orleans didn't have to hide anything from anyone, really, and most of them actually operated out of the bayous, not the city.
New Orleans only has one tunnel, and that isn't even in the city proper, but out in Gretna, if memory serves, and is only a block long.