The plane, which was carrying about 500 people, landed at JFK at about 9.10am local time (2.10pm BST).
A spokeswoman for the airline said 10 passengers and crew were taken to hospital. Reports had earlier put the number of unwell people at about 100.
The sick passengers reportedly suffered from “fevers of over 100 degrees and coughing”.
Passenger Erin Sykes said on Twitter: "Worst flight ever Dubai to NYC; plane was basically a flying infirmary.
"Many of these people should never have been allowed to board. Now sitting on the ground after 14 hour flight unable to leave."
Larry Cohen, who was also on board, posted photos on Twitter showing dozens of police and emergency vehicles waiting outside the plane on the tarmac.
An Emirates spokeswoman said: “All passengers were screened by the local health authorities prior to disembarkation and three passengers and seven crew were transferred to the hospital for further medical care and evaluation.
“Nine passengers underwent additional medical screening at the site near the aircraft and were released afterwards.
“The rest of the passengers were allowed to leave and clear customs."
A spokesman for New York City mayor Bill de Blasio said representatives of the US Centres for Disease Control and Prevention were at the scene.
A New York Police counter-terrorism division tweeted that it was monitoring what appeared to be a "medical situation".
