Mexican President Claudia Sheinbaum also expressed sadness on social media at the "loss of two crew members of the Cuauhtemoc Training Ship."
The tops of the Cuauhtemoc’s three masts struck the city's landmark bridge, partially collapsing as the vessel moved along the East River.
Adams said the 142-year-old Brooklyn Bridge was spared major damage.
The New York transport department posted on X that there were no signs of any structural damage to the bridge.
The cause of the collision is under investigation.
What else do we know about the crash?
Multiple eyewitness videos captured the masts snapping and partially collapsing as they crashed into the bridge's deck. Videos showed heavy traffic on the bridge at the time of the collision.
The vessel then drifted toward the river's edge as onlookers scrambled away from the shore. Sailors could be seen aloft in the rigging on the damaged masts.
The New York Post reported that victims were rushed to local hospitals.
Nick Corso, a witness, told AFP that it looked like there was a lot of panic aboard the vessel with "lots of screaming" and "some sailors hanging from the masts."
The Cuauhtemoc, a barque built in 1982 with a mast height of 48.2 meters (158 feet) had been on a training maneuver at the time and was damaged in the "mishap," a statement posted by the Mexican Navy on X said.
