The Alexandria school system does not have the space, the money or the responsibility to help solve the problem of teen-age pregnancies by setting up a health clinic at T.C. Williams High School, School Board Chairman Lou Cook said last night.

The proposal for a comprehensive health clinic in or near T.C. Williams, the city's only high school, was one of several recommendations of a United Way and Alexandria Youth Services Commission task force on adolescent pregnancy. According to statistics compiled by Planned Parenthood of Metropolitan Washington, 360 Alexandria teen-agers became pregnant in 1983, giving the city the highest rate of teen pregnancy in Northern Virginia in that year.

Although no vote was taken at last night's School Board meeting, most board members seemed to agree with Cook that such a clinic is beyond the resources and the responsibility of the schools.

Cook estimated that a comprehensive health clinic would cost between $100 and $125 for each of the 2,400 students at the school. "We don't have the cash to do that," she said. "We do not have room in T.C. Williams for anything else at this point. Also, how far is the school supposed to go in solving society's problems?"

Last week the Annandale Women's Center, a private state-regulated facility at 2817 Duke St., announced it would provide free birth control devices, counseling, treatment of sexually transmitted diseases and pregnancy tests to all Northern Virginia high school students who request them beginning next Tuesday.

Gail Frances, president of the center, said the facility decided to pay for those services because local governments and school systems have backed away from the issue.

The Alexandria task force's report, completed last December, said access to contraception is of "primary importance in the prevention of teen pregnancy." School-based health clinics in several communities around the country "provide a host of necessary adolescent health services including facilitating the use of birth control and thereby reducing pregnancy and repeat pregnancy," the report said.

Even if the school system had enough money and space for a school-based clinic, Cook said she doubted teens would use a health center geared exclusively toward sex counseling and birth control because they would fear being teased by their peers. "Kids will not walk through a door that says 'Birth Control.' If the sign says 'Hang Nails, Etc.,' they might," she said.

Cook said the existing Family Life Education curriculum aims to reduce teen pregnancies through education. "One of the things we're trying to do in [those courses] is teach them about the responsibilities of becoming a parent, which are difficult and onerous," she said.

The task force concluded that Alexandria teen parents particularly need affordable child care, programs that stress the completion of formal education and job training programs. The group recommended, among other things, classes to educate parents about teen sexual behavior and to teach parenthood skills to teen-age mothers and fathers.

The school board approved an application last night for a $4,000 March of Dimes grant to finance a sex education course for parents and parent-child workshops designed to improve discussion about sexuality.

Three such workshops -- one for fourth through sixth graders, one for junior high and one for high school students -- would include lectures on male and female reproductive anatomy, filmstrips, games and discussions "to support the concept of family focus sex education," according to the grant proposal.