The Fairfax County School Board last night approved a $586 million budget proposal for the next school year after overwhelmingly rejecting an attempt to add more money for teacher pay than the superintendent proposed.

The budget for the area's largest school system would rise $49 million over this year's if the Board of Supervisors approves the spending request. The supervisors vote on their budget in April.

"We did as much as possible to transfer resources into the classroom," Board Chairman Mary E. Collier said after the budget vote.

The board approved the budget 8 to 1, with member Katherine Hanley opposed and Anthony Lane absent, after a 3 1/2-hour meeting at the Luther Jackson Intermediate School in Falls Church, whose auditorium was jammed with more than 800 people. Many were teachers, and wore buttons that read, "We want what's right, not what's left."

The audience booed heartily when the board, voting 7 to 2, rejected a proposal by Hanley, who represents the Providence District, to add $250 a year to the pay of teachers in the middle range of the salary scale.

Critics of Superintendent Robert R. Spillane's teacher-pay proposal said it would help teachers at the top and bottom of the salary range, but do nothing beyond cost-of-living and grade raises for those in the middle.

"This is not a great deal," Hanley said, "but it's a step in the right direction." Board members opposing additional money for teachers have said they want to wait for a report from Spillane's newly appointed commission that is studying the issue of appropriate teacher compensation, including merit pay.

Only Hanley and Annandale District member Laura I. McDowall voted for the motion, which would have made cuts in other areas to pay for the added teacher wages.

Immediately after the board's final budget vote, teacher Donna Caudill, president of the Fairfax Education Association, and a crowd of members of the teacher organization walked out of the meeting in protest.

Caudill said teachers are full of "rage and frustration" at the School Board's vote to "turn its back on the thousands of taxpayers who called for improved salaries." County teachers have been protesting through a job slowdown for almost three months.

She said her group will organize a massive lobbying campaign to persuade the Board of Supervisors to reverse the School Board vote against more money.

The budget includes a 4 percent cost-of-living raise for all school employes, including the 7,300 teachers, and additional money for the newest and most veteran teachers.

Beginning teachers, now paid $18,385, would receive $20,000. The average teacher's salary in the county is about $29,000.

The budget also would include:

*Additional guidance counselors for intermediate and elementary schools.

*$1.4 million for classroom computers and the staff to run them, and $1.1 million for computer and other equipment for school offices.

*A reduction in the maximum kindergarten class size, from 30 to 28. In 22 schools with "special needs," such as a large number of minority or foreign-born students, the maximum would be 24.

*$1 million for additional teachers and clerks at eight intermediate and nine high schools with special needs.

*An additional reading teacher and librarian for the county's largest elementary schools.

*$412,960 to hire 13 elementary school art teachers as part of a program to provide an hour a week of art instruction to all elementary school pupils by the end of 1990.