The D.C. public school system plans to dramatically expand summer school programs this year to accommodate more than 13,000 students--the most ever--and for the first time to offer courses for gifted children in addition to the traditional remedial courses.

D.C. school officials said that $1,879,508 has been earmarked for the summer session beginning in late June, an increase of 61 percent over last summer's $1.17 million.

In the past, summer school in the District has generally involved only remedial work. This year, however, the school system will offer various enrichment courses, including a computer camp and 21 three-week courses for talented students, as well as an expansion of remedial courses.

In suburban Maryland and in Northern Virginia, school districts either are offering a modest expansion in summer-school programs or are simply maintaining last year's levels, officials said. Almost all, however, have responded to the increasing interest in computers by offering summer courses on their use and applications.

In the District, a major reason for the ability to expand summer programs is the size of the current overall school budget, which increased from $272 million in fiscal 1982 to the current year's $306 million. Only two summers ago, in 1981, the D.C. school system canceled its entire summer program because of widespread budget cuts.

About 9,000 students attended last year's D.C. summer school program, officials said. This summer, $1.1 million--a figure nearly equal to the entire summer program budget last year--will be used to offer remedial help in the basic English, mathematics, science and social studies courses offered during the regular school year.

School system spokeswoman Janis Cromer said that some of the summer programs for gifted students include one on Shakespeare, given in connection with the Folger Library, in which 15 students study and act in the classic plays; a course on critical thinking and problem solving, and work at radio station WPFW.

Ninety D.C. students will also spend a week learning about computers at Camp Broad Meadow in Maryland, and there will be eight sites in the city where 80 other students in grades 4 through 9 can take two-week computer courses.

Summer school is free in the District, except for some small costs for materials. Most other Washington-area jurisdictions charge tuition for summer classes, ranging from about $32 a credit to $158 a credit in Prince George's County.

"We tried to make the courses self-sustaining last year, and now it is as close as we can make it," said Prince George's school spokesman Ellie Rotter. The county this year will cut the number of summer school centers from 15 to 12 and expects 8,000 students this summer, down from 10,158 two summers ago. "We are trying to hold the line as close as we can."

Montgomery County officials said their summer school budget this year is roughly comparable to last year's $1.2 million. Montgomery is expanding its elementary-level computer course and offering a two-week French camp for the first time.

Fairfax County is trying to make its $2.7 million summer program more self-supporting by charging full tuition for all students except those with financial hardships, said Max Skidmore, director of special programs and services. In the past, tuition has been reduced for many students to encourage attendance.

Alexandria's summer school budget will increase from $164,335 last year to $182,871 as the school system expands its enrichment courses in science and computers. "In previous years, we didn't offer anything except remedial courses," school spokesman Mel Alba said.

The Arlington County schools expect about 3,000 students this summer, a slight increase over last year. For the first time, the county will be offering a computer course, associate superintendent Webb Boyd said.