"I don't think you should cut education, because it's our future," 16-year-old Dina Schiodt told AFP in Copenhagen's central City Hall Square.
"If we have bad courses things will go downhill in the future because we won't have the right skills," she added.
Copenhagen police said between 10,000 and 15,000 had taken to the streets in the capital, while half that number demonstrated in Denmark's second largest city, Aarhus, according to Danish daily BT.
The government has said it needs to cut the budget for upper secondary education and universities by two percent each year over the next four years, arguing that the country already spends more on education per capita than any other nation.
Critics say it would result in one in ten teachers losing their jobs, while the government has insisted that savings can be made without affecting the quality of education.
Denmark was the OECD country that spent the most of its wealth on education in 2011, the latest year for which data was available, the organisation said last year.
See also: Denmark spends most on education: OECD
The Scandinavian country spent 7.9 percent of its gross domestic product on education followed by Iceland at 7.7 percent and South Korea at 7.6 percent.
Danish teachers' wages were higher than the OECD average, with starting salaries 40 to 50 percent higher than the average, it said.