Not if you are a US citizen flying directly to and from Puerto Rico.
Even newborns need passports to enter other countries. Even Puerto Rico and Mexico now.
A Passport is not required if you are travelling directly between the United States and Puerto Rico.
As long as you are traveling directly from Puerto Rico to NY, without stopping in any foreign countries - for example, elsewhere in the Caribbean or in Canada - you do not need a passport to travel. Puerto Rico is a US Commonwealth. Travel to or from Puerto Rico and/or the US Virgin Islands to the US mainland is travel within the United States and passports are not required. You will, however, need a government issued photo ID, such as a drivers license.
Plane
No.
Puerto rico is a terretory of the United States, you do not need a passport to go there.
You do not need any special documents such as a passport to go to Puerto Rico because it is considered a US territory. All you need is some sort of identification.
Not from United States.
No. No passports are checked when flying on a direct flight from Puerto Rico to a US Airport. However one must have at least one valid state-issued ID like a driver's license to fly anywhere from any US airport.
Do you need a passport to travel from the U.S. to Puerto Rico, then Puerto Rico to U.S.?
To go to Puerto Rico is like to go to any other state. You'd need the same things you would need to go from Colorado to Utah.
Puerto Rico is United States territory. No passport is necessary if you are a US citizen. The same applies to the U.S. Virgin Islands. You still must have valid identification and be taking a direct flight. If you were to change planes in Mexico, for example, a passport would be required. If you are not a US Citizen or Green Card Holder, you would need a passport to enter the US in the first place but to go from Puerto Rico to other parts of the US is domestic travel and a passport is not required. However a passport can be the government-issued photo ID that is required for anyone to to board a commercial airplane.