Carlisle City Council said it would have cost about £26,500 a month to keep the site operational, which it could not afford.
Campaign group Friends of Carlisle Victorian and Turkish Baths said the fight to keep them open would continue.
The council said the Sands Centre redevelopment would include a 25m main pool and a 20m learner pool complete with an moveable floor.
The old Victorian baths on the site opened in 1884 and the Turkish baths were added in 1909. The building has Grade II-listed status and cannot be demolished.
Julie Minns, chair of the Friends of Carlisle Victorian and Turkish Baths, said the campaign would keep going.
She said: "Our focus from the outset has been to keep the Turkish baths in use and it's deeply disappointing the council voted to prematurely close what are the North West area's last Turkish baths.
"We hope to persuade the new Cumberland Council to reopen them next year and in the meantime we are progressing our plans for a community asset transfer and redevelopment of the baths as a new centre for health and wellbeing."
Carlisle City Council leader John Mallinson, said: "I'd like to thank all those involved in driving the Sands Centre redevelopment forward during challenging times.
"It represents a cornerstone in our plans to support our communities by improving their health and wellbeing."