The council is to stop directly providing the services to save £2.8m in the next financial year, rising to £4.5m a year until 2021.
More than 16,000 people have signed a petition against the cuts.
A group of campaigners protested outside the council meeting at which the savings were approved.
The authority said its research showed only one out of every five young people in Staffordshire had used a county council-funded youth facility.
Cabinet member for community safety, Mark Sutton, said the authority wanted to focus more on "community-based activities".
"Young people simply no longer want to attend a traditional youth club," he said.
"This fresh approach is about improving services in the community and increasing access to schemes such as the Duke of Edinburgh which really support young people's aspirations around education and jobs."
The clubs could be closed down or run by other organisations, the council said.
The Unite union said 50 full-time and 350 part-time workers could be made redundant.
Protester Alex Hilton said: "The impact for young people will be detrimental.
"This service has helped me and many of my friends find work and helped with other issues whether through youth clubs or one-on-one counselling.
"It's a real shame because we're now going to see a limit put on how people grow up and move into the working world."
The council said it needed to save £109m from its overall budget over the next five years.
More than 30 youth clubs in Staffordshire could close with the loss of 400 jobs after county councillors approved cuts to youth services.