Our first Cunard trip (2015) was a transatlantic from New York to Southampton.
We flew to New York about three days before the trip. Booked own flights to JFK, taxi to Distrikt Hotel, 342 West 40th Street. Great central location in NY, and much cheaper than booking via Cunard. A friend had previously stayed there, and we would have no hesitation in recommending it.
Hotel booked us a taxi to Red Hook at Brooklyn.