The Fairfax County Board of Supervisors yesterday adopted a budget that will cost the average homeowner approximately $150 more in real estate taxes next year, despite a three-cent cut in the country's estate tax rate.

Although the $511.3 million budget represents an increase of $59.3 million over the current spending level, county officials said it will probably force cuts in payroll and some programs. While education appeared the hardest hit by the cuts, county police gained an added $1 million -- money the supervisors said was needed to bolster the number of officers on the 722-member force.

The board voted the additional police money after Supervisor Marie Travesky (R-Springfield) said many Fairfax residents are upset over the level of police spending in the county, the lowest on a per capita basis of any suburban jurisdiction in the Washington area. "We have a very serious problem with not enough officers on surveillance, not enough officers to deal with drug and controlled substance abuse in our neighborhoods," Travesky said. "If there is one recurring theme in my letters and phone calls, it is that we need more police officers."

Most of the increased Fairfax spending next year, however, will go to counteract inflation, Fairfax officials said, including the cost of a 9.1 percent cost-of-living increase for employes, and higher costs for energy and transportation.

County Executive J. Hamilton Lambert said the work force reductions will probably come about through attrition. "I do not anticipate any layoffs," he said.

The county's schools lost $3.6 million from their proposed budget of $394.7 million. Fairfax School Superintendent L. Linton Deck criticized the cuts as "out of kilter," noting that the schools may suffer further from the Reagan administration's planned reduction in federal aid.

Deck declined to speculate on where the cuts would be made. "We will have to make some difficult decisions," he said.

Approving the budget after almost five hours of debate, the supervisors unanimously adopted a tax relief motion that will reduce the county's tax rate from $1.54 to $1.51 per $100 of assessed valuation beginning July 1. "I think we ought to start thinking about what the citizens can afford to pay -- not what programs we want," said Supervisor Thomas M. Davis III (R-Mason), who sponsored the tax rate cut.

But with county housing assessments climbing by almost 15 percent over the last year, the three-cent reduction does not even come close to actually cutting the average homeowner's tax bill. Under the new tax rate, a median-priced home valued at $89,400 would carry a tax bill of approximately $1,350. The same house, which last year would have been valued at $77,800, would then have had a tax bill of $1,198.

Supervisor Sandra L. Duckworth (D-Mt. Vernon) had urged the board to adopt a tax cut of four cents by imposing sharper spending cuts, but the supervisors rejected her proposal after hearing Assistant County Executive J. Patrick McDonald warn that future revenues are threatened by the nation's "precipitious economic uncertainty."

With mortgage rates hovering near 16 percent, McDonald said, the country cannot expect continued, sharp increases in real estate, sales and personal property taxes.

"It isn't a prudent action to cut the tax rate to the point where we would have to raise it again next year," said Supervisor Martha V. Pennino (D-Centreville), before the board defeated Duckworth's measure 6 to 3. Joining Duckworth were Republicans Davis and John F. Herrity, the board chairman.

Several supervisors criticized school officials for the size of their proposed budget, which had sought an increase of 18.5 percent in county support and included plans for some $3.5 million in program improvements. "I just think that in an aging county, with the population going up, their priorities are all screwed up," said Davis. "What they have to realize is that the constituency for schools is declining."

Supervisor James Scott (D-Providence), who voted with the rest of the board to cut the education budget by $3.6 million, later expressed doubts about the fairness of the cut. "I view it as a really stiff cut in the education budget, much larger than I think we can afford," he said. "It really can cut into the muscle and probably the bone of the school system."