The Fairfax County Board of Supervisors, responding to a record-breaking burst of construction, approved yesterday the addition of 77 permanent staff members to the swamped county agency responsible for inspecting new buildings.

The 26 percent expansion of the 291-member Department of Environmental Management is budgeted at $3 million in the coming fiscal year and is estimated $2.4 million in each future year. Increased fees to builders and developers will pay almost all of the cost.

The 1,683 building permits issued in January were 52 percent more than the 1,106 issued in January 1984, the year in which Fairfax County set an all-time record for the number of building permits filed.

"I have been here 20 years and I have never seen an . . . increase like this," County Executive J. Hamilton Lambert told the Board of Supervisors yesterday.

"It is a crisis," Supervisor Audrey Moore (D-Annandale) said, adding that the agency responsible for enforcing the county building code is "so overburdened you would not believe it."

Claude Cooper, the agency's director, said inspections are running days behind schedule, delaying construction, and inspectors are spending less time at each site, making more cursory inspections in order to get to as many sites as possible.

The board also approved funding for 20 temporary inspection positions for the peak spring and summer building season.

In addition, county staff was ordered to develop a plan for contracting out some inspection chores to private engineering firms for those times of the year when the workload is too heavy for the staff.

The agency's original request, made last year, was for 210 more staff members. Although it was trimmed to 77 by yesterday, it still prompted Board Chairman John F. Herrity to say: "There is no question we need to staff up, but there is a very basic issue of how many permanent people we need. I have a very serious concern about building up a big staff bureaucracy here." Herrity joined the other supervisors in a unanimous vote for the increase.

Of the 77 new positions, 34 will be inspectors; the remainder will be technical and support personnel.

Cooper also said modified inspection procedures might be necessary if the burden on his staff is not eased.

Those include discontinuation of "home improvement-type inspections such as reroofing . . . . devoting relatively less time to drainage and general complaints" and "reducing the target response time from one day to three or . . . as many as five days."

Those proposals were greeted with frowns and denunciation by board members, who directed Cooper not to implement any without coming back before them for prior approval.