Most of the children suffered minor injuries and were treated at the Oyster Festival in Norwalk, police said. But several others were more seriously hurt and taken to local hospitals, according to authorities.
One hospital reported three in stable condition and another reported three were being evaluated. Victims were also reportedly taken to a third hospital.
It wasn't clear how far off the ground they were when they fell.
Norwalk police called in the state fire marshal's office to investigate.
The festival's organizer, the nonprofit Norwalk Seaport Association, said it directed the ride's operator, Stewart Amusement, to shut down the entire ride area until state inspectors complete an inspection. The other rides later reopened and the rest of the festival remained open on its third and final day.
Stewart Amusement didn't immediately return phone and email messages seeking comment. Its rides are inspected by its own staff every day, by state and local inspectors weekly and by engineers and insurance inspectors each year, the company's website said.
"Your safety is of critical importance to us," the site says. "Not only do we have an obligation to provide our guests with the safest equipment and environment possible, but also our ultimate success depends on it."
Stewart Amusement says it has provided rides and other attractions since 1983 at events in Fairfield and New Haven counties in Connecticut and neighboring Westchester and Putnam counties in New York.
Troopers with state police fire and explosion unit were investigating, department spokesman Lt. J. Paul Vance said.
Festival organizers posted a statement about the accident on its Facebook page.
"Our first and only concern is for the wellbeing of those involved and their families," the statement said.