According a news article in today's
Daily News, Archbishop Dolan is petitioning the Holy See to name
Old Saint Patrick's Cathedral a minor
basilica (the
Wiki article as some other useful info). There are 2 basilicas in the Brooklyn Diocese but none in the Archdiocese. While it is largely an honorary distinction, it is seen as important and therefore prestigious. The old cathedral is quite a place as it continues to serve the spiritual needs of God's people. For example, Holy Mass is celebrated in English, Spanish and Chinese each week.
There are three distinctions for basilicas: there are four major basilica, directly connected with the Pope, and therefore are called "papal" (St Peter's, St Paul outside the Walls, St John Lateran and St Mary Major); there are eight pontifical minor basilicas (all in Italy); and the rest of the world's basilicas are "minor" basilicas but no less distinguished because they too are connected with the Holy Father but they don't have "papal altars and thrones."
In the USA there are 60+ minor basilicas. If the Holy See makes Old Saint Patrick's a basilica it would take precedence over the other NY churches except for the Cathedral, have the use of the tintinabulum (a bell on a pole which would signal the arrival of the pope) and an umbrella (carried over the
pope for protection) and carried by the laity behind the processional cross; the rector would be able to wear a grey mozzetta. The list of basilicas can be read
here.