Public school enrollment continued to tumble this fall throughout the Washington area, increasing the pressures to close school buildings and hold down school spending despite strong protests from parents.

The drop was steepest in the District of Columbia where enrollment plummeted almost 8 percent from a year ago. The decline areawide was 4.6 percent.

In both the city and the area it was the sharpest drop yet in a decade of falling enrollments, caused by declining births and the outward movement of families, who often are replaced by singles and childless couples.

"The baby-bust is taking its toll," said Atlee Shidler, executive vice president of the Center for Metropolitan and Municipal Research. "And an awful lot of families with children are leaving the metropolitan area."

In Arlington, the enrollment drop was almost as great as in Washington -- 7 percent, while declines of just under 5 percent were reported in both Prince George's and Montgomery counties.

Even counties at the edge of the metropolitan area, where school enrollment grew substantially during the first half of the 1970s, lost students this year.

For example, both Howard and Charles counties in Maryland had slight enrollment declines for the first time in more than a decade.

Of the counties on the Capital Beltway, the only one with a relatively modest enrollment decline was Fairfax. The number of students in its public schools fell by 1.1 percent this year to 128,063. This allowed Fairfax to become the largest school system in the area for the first time, topping Prince George's county by about 900 students.

The decline in school enrollment in the Washington area is part of a nationwide trend, stemming from the sharp drop in births during the early 1970s. But the overrall decline here is much steeper than it has been throughtout the country.

"The houses are still springing up around here," said Joyce Harte, supervisor of community relations for the Prince William County school system, which has lost about 2,500 students over the last two years. But when the people move in, they have no children or very few children, and the enrollment here goes down."

Since 1970, overall school enrollment in Washington and its close-in surburban jurisdictions has dropped by 119,400 students to 490,200 this fall, according to figures complied by the metropoolitan research center and local school districts.

The decline of 19.6 percent compares to a 9.5 percent public school enrollment drop nationwide over the same period.

This fall's decline here was the greatest yet, a drop of 23,400 or 4.6 percent, compared to a nationwide decline of about 2 percent according to the National Center for Education Statistics.

No areawide enrollment figures are available for private schools, although demand for private schooling apparently is strong. Thoughout the country, private school enrollment has been steady at about 5 million students since 1972 despite a loss of 4 million students in public schools. Even so, about 89 percent of American children attend public schools. In Washington, where a mid-September enrollment report shows a drop of 8,789 from the same time last year, Mayor Marion Barry has proposed a 4 percent cut in the 1981 school operating budget.

Barry said the cut is needed to free money for other city programs and to avoid tax increases. He said the Board of Education could save the money by making the "tough decisions" to close underused buildings and cut the number of teachers and other employes in proporation to the enrollment decline. n

In Arlington and Montgomery counties, school superintendents already have made proposals to close more schools next June.

Virginia Gov. John N. Dalton has said the enrollment drop should allow jurisdictions in Northern Virginia to use more local tax revenue for Metro transit operations rather than receiving a state subsidy.

On the other hand, teacher and parent group have argued that the enrollment drop should be used as an opportunity to lower class size, provide more special programs, especially for handicapped children, and expand classes beyond the usual 5 to 18-year range, with public nursey schools and more classes for adults.

In Washington, where enrollment has dropped by 30 percent from its peak in 1969, officials said the decline reflects the growing migration of black familes to the surburbs as well as the decrease in births.

Alexandria, which has undertaken a major busing program for desegregation, has lost 59 percent of its white enrollment since 1970. However, this fall for the third year in a row, the number of black students declined at about the same rate as the number of whites -- 3.5 percent. The racial balance of the school system was virtually unchanged -- 47 percent black, 44 percent white, and 9 percent Asian and Hispanic.

This fall's drop in Prince George's County was the steepest since 1973, the year court-ordered busing began. But officials said data has not yet been compiled on the racial composition of the school system this year.

In Montgomery County, all the decline was among white students. The number of blacks and Asians increased by about 400 each, as minority-group enrollment reached 19.5 percent.