Washington's summer jobs program starts in three weeks, and Mayor Marion Barry, who is attempting to get the program off to a good start in an election year, is already facing criticism from his political opponents who don't like the way the program is shaping up.

The summer jobs program for several years has been a political embarrassment to Barry, who promised in the 1978 mayoral campaign to create 30,000 summer youth jobs--twice the number during the Walter Washington administration. But Barry's administration has never reached that level of job production, although he came close with about 28,000 jobs his first year in office. That was when criticism for mismanagement of the program was most shrill.

In succeeding years, the program was criticized for a number of management problems, from late paychecks to poorly supervised work sites where some youngsters were paid for doing almost nothing. Again, as in years past, some of the mayor's critics are complaining about the kinds of jobs available for youth and what they now charge is election year maneuvering.

"There are 17,000 youth jobs this summer," said City Council member and mayoral candidate Betty Ann Kane in a prayer breakfast speech two weeks ago. "Do you know how many are in the private sector? Only 300. It does youth a disservice to only have work in playground recreation centers."

Ivanhoe Donaldson, former head of the city's Department of Employment Services, said Kane is correct that the Metropolitan Washington Board of Trade and the National Alliance of Businessmen have been asked to find only 300 jobs for Washington teen-agers.

But Donaldson, who left the department last week to take over Mayor Marion Barry's reelection campaign, said the city is trying to locate another 700 jobs with private employers for job-hunting teens.

"We can't use the CETA funds to subsidize jobs in the private sector," said Donaldson, referring to money the city receives from the federal Comprehensiuve Employment Training Act, "and we don't want to use city funds for too many jobs with private employers. That would tip the scales in the market, . . . help only one guy to run his company and maybe not his competitor."

According to Matthew Shannon, who is replacing Donaldson as head of the employment department, 3,800 of the 17,000 jobs will be in city government, with about 500 of those in the recreation department, 800 in the Department of Human Services and 1,000 working on building maintenance and as office aides in the school system.

"To have that many jobs in city government shows a lack of imagination," Kane said in an interview. "It's much harder to develop jobs in the private sector, to say to stores and other businesses, 'We want you to hire one, two or three young people and we will provide the rules.' "

A little more than half the summer jobs, however, are not in city government at all but in nonprofit groups, which will have 9,000, according to Donaldson. The largest of these nonprofit groups is Associates for Renewal in Education, which will employ 4,000 14- and 15-year-olds to tutor first-, second- and third-grade children. The remaining jobs with nonprofit organizations will be with groups such as the Urban League, Southeast House and Arts D.C.

Donaldson denied that any of the decisions about which nonprofit groups would get young workers were made on a political basis in this election year, and he added that with the exception of a few computer problems--such as sending some job notices to the wrong addresses--this year's program is doing "fantastic."

In a recent interview, Barry said he had "overloaded the system" trying to reach the goal set in the 1978 campaign and maintained that sending Donaldson to head the employment department and having fewer youngsters in the program have ironed out many of the difficulties. But the program's crippling problems in the first two years under Barry have not been forgotten by his opponents in the present mayor's race.

"What he Barry did to the young people of this city is a shame," Charlene Drew Jarvis, Ward 4 council member and a candidate for mayor, said in an interview. "He promised all those jobs, didn't deliver them and the extra jobs he put out there weren't organized or supervised, so the kids got the message you can get paid for doing nothing and the employers don't expect anything of you."

Of this year's 17,000 jobs, 8,500 will be paid for with $7.4 million from the CETA program (which is scheduled to be eliminated by the Reagan administration) and the other 7,500 with $5.1 million in city government funds. The city part of the program has no income requirements for the young people who apply, and Barry's opponents in the mayor's race have been grumbling privately that many of those jobs may be used to curry support for the mayor.

But Donaldson said most of the jobs were filled on a first-come, first-served basis after youngsters began applying for the jobs in December, and the remaining jobs are being allocated on a priority basis to the handicapped, single parents, Hispanos, children referred by the courts or social service programs and high school dropouts.

Still, suspicion that the program will be used as an extension of the mayor's reelection effort persists.