The Fairfax County Board of Supervisors, fearing the county may face harder times from the recession, yesterday approved a fiscal 1983 budget that will raise the average homeowner's real estate taxes by $102.

The board cut the tax rate from $1.51 per $100 of assessed property to $1.47, while raising the budget from about $520 million to $564 million. Because assessments rose 10.5 percent, increasing the average home value to $98,848, the average tax bill will rise 7.6 percent from $1,351 to $1,453.

The supervisors said they were unwilling to reduce the tax rate further because they expect tax and fee collections to slump next year as a result of the recession and the housing industry's collapse. The board will be up for reelection then and property taxes are likely to be a campaign issue.

"I think most people want us to act responsibly; that we not make cuts this year that will come back to haunt us next year," said Springfield District Supervisor Marie Travesky.

The Federation of Citizens Associations and the Chamber of Commerce both urged a greater reduction in the tax rate, but the supervisors found little to cut in the budget for the year beginning July 1 that was proposed by their appointed county executive. The tax rate is subject to a second vote next week, but such votes typically are a formality.

Under the budget approved yesterday Fairfax employes will receive a 5 percent cost-of-living raise, with some workers receiving an additional 5 percent merit raise, and most county programs simply will be continued at their current level of funding.

The supervisors capped months of debate with their appointed school board by cutting $4.3 million from the proposed $429 million school budget. The supervisors, arguing that the school department is sometimes extravagant and top-heavy with administrators, said they would cut more if they could control where reductions are made. The supervisors will give the school system $280 million, or two-thirds of its total budget, but under Virginia law the supervisors have no line-by-line control over school spending.

"It's going to hurt," said school Superintendent Linton Deck. "I have no idea where the cuts will come from."

Several school administrators said the cuts probably will come from proposed programs, which the supervisors criticized as inappropriate. Those proposed new programs include computer training for intermediate and high school pupils, new equipment and supplies and more teacher aides for students with learning or language problems.

The supervisors created several reserve accounts after their budget chief reported the county should earn about $13 million more in 1982 and 1983 than he estimated previously. Fairfax budget chief James P. McDonald said small cars have retained their value more than anticipated, leading to increased personal property tax collections, and the county grew faster than expected before the housing slump.

The supervisors set aside $7.8 million of the windfall to settle a discrimination lawsuit brought by the Justice Department on behalf of county employes and job applicants and to help fund the controversial Dulles Toll Road. The board also set aside $20.2 million to protect against declining revenues in future years.

Federal aid to Fairfax declined by about $8.7 million this year because of budget cutbacks and county officials said their share of federal funds may drop between $7.0 and $7.5 million next year. More uncertain is the effect of the recession, which has hit hard two important sources of county revenue, car sales and housing construction. McDonald predicted a county deficit of more than $17 million at current tax and spending rates by fiscal year 1985 even if the recession ends this year.

"We have no idea when the economy's going to turn around and we have no idea what's going to happen to the federal budget," said County Executive J. Hamilton Lambert. "The prudent thing was to establish a reserve."