Oxfordshire County Council plans to close most of its 44 children's centres and seven other units to save £8m.
Eight remaining centres would be used to form part of a new referral service for babies to 19-year-olds.
Under its preferred option, the council also proposes to stop providing "universal" services such as stay-and-play or open access youth sessions.
About 70 people have been protesting outside County Hall over the plans.
Mother-of-twins Marion Reece, from Oxford, said the centres have provided her with "crucial" support.
"They provide support and a service that is very difficult to quantify, and I think it's only when that's taken away that it will be obvious how they have supported people and prevented other problems from developing," she said.
On BBC Radio Oxford's Facebook page, Chloe Dallimore described news of the plans as "a tragedy".
Maureen Norton described proposals as "very short-sighted as families will need support from somewhere - where will they go?"
"Our looked-after children population is going exponentially up," said Councillor Melinda Tilley, cabinet member for children.
"We need to look at how we deliver the service for the most vulnerable.
"We've been looking at a completely new service so that we can merge all our social work teams."
Dave Ricketts, from the union Unite, which represents staff at the children's centres, described the proposed closures as "depressing".
"To find most of these staff being potentially laid off and made redundant is obviously a very devastating impact on our members and even non-members who work for the authority," he said.
He said central government had failed to provide sufficient funds to run the services.
The locations for the eight centres have not been finalised but are expected to be in Banbury, Bicester, Witney, Abingdon, Didcot, Blackbird Leys, Rose Hill/Littlemore and Barton/Sandhills.
In October 2013, fears that some children's centres could close led to led to protests.
Cuts of £3m to the early intervention service were agreed in 2014, with a further £3m in February 2015.
An additional £2m has been added in anticipation of the chancellor's autumn statement.
A consultation is expected to be held between September and December 2015.