One of the babies is three days old and suffered a stab wound to the stomach, another is 20 days old and suffered lacerations to the ear, chin and lip.
The ages of the infants, two girls and a boy, were not clear, but one of the girls is in serious condition, according to police.
The woman, 52, has been taken to hospital under police custody with non-life threatening injuries.
Police said the suspect was in custody and taken to a local hospital after trying to kill herself.
The two adults were a co-worker at the facility and the father of one of the infants, the officer said.
The incident occurred at the daycare center in the borough of Queens at about 3:30 a.m. EDT, a spokesman for New York City Police Department said.
Police arrived and discovered the woman in the basement kitchen with self-inflicted knife wounds on her arm.
A motive hadn't been determined.