Madison Square Garden Co., whose venues include New York City’s Madison Square Garden and Radio City Music Hall, said hackers may have captured personal information on credit and debit cards used at some of its venues over the span of nearly a year.

The company said cards whose data may have been captured were swiped in person at food and beverage locations from Nov. 9, 2015, through Oct. 24 of this year at Madison Square Garden, the Theater at Madison Square Garden, Radio City Music Hall, the Beacon Theatre in New York and...