It was one of those things that politicians frequently do to break the ice. But it didn't work. As a matter of fact, it backfired.

Mayor Marion Barry was sitting at a table with a group of potentially hostile leaders of the District's trash collection union.

"You know," Barry said, matter-of-factly, "My stepfather helped organize the (trash collectors) union in Memphis."

The union leaders barely reacted. In fact, one of them, Geraldine Boykin, appeared especially unimpressed.

"You need to talk to your stepfather more," she bluntly told the mayor -- and she didn't smile.

What she meant, Boykin told a reporter later, was that Barry's proposed 1981 budget indicated little compassion for the plight of working men like those in her union, District Council 20 of the American Federation of State, County and Municipal Employes. In other words, people in unions like the one Barry's stepfather helped organize.

"The mayor should know better. He's lived in this city. He's been a worker," Boykin said.

"At one point during the mayor's administration, I thought we were beginning a good sound labor-management relations. Now, I don't know," she said. "I get kind of nervous. He's using things that have been used throughout the country to intimidate the work force. If that happens, there won't be any labor-management relations. We'll just declare war."

Next year, for the first time, the District government will have full-fledged collective bargaining on working conditions and wages with more than 40 labor groups. Boykin, whose union represents 14,000 of the city's 35,000 regular employes, is not the only one worried.

Donald M. MacIntyre, regional vice president of the American Federation of Government Employes, said the Barry budget signals a clear intention to give private contractors much of the work now done by city employes.

"Our experience in the federal sector is that's bad news," said MacIntyre, whose union represents about 8,500 city employes. "You've seen what happened (with contracting) with GSA (the General Services Administration). No one benefits except the people who get the payoffs and the kickbacks."

William H. Simons, president of the Washington Teachers Union, says the initial indications in the Barry budget, including a proposed $10 million cut in school spending that would eliminate 839 jobs, are troubling.

"If I take the '81 budget on its face," Simons said, "I would say that the situation would be dismal." His union represents about 5,000 teachers.

Barry was rudely introduced to the volatile world of municipal labor relations last spring when the teachers union staged a 23-day strike that sent the mayor scrambling for a way to end the strike and head off a full-scale confrontation with organized labor in Washington. He could receive another lesson next year.

This week, Barry withdrew a controversial plan to implement once-a-week trash collections in some parts of the city. That momentarily ended his problems with a AFSCME, but left unaswered still other pressing questions Boykin and others have about trash contracting proposals that would severely reduce the environmental services payroll.

The mayor onsiders it part of his mandate to cut city spending and hold down taxes, while at the same time not diminishing the quality of city services. One way to do that, for example, is to contract work.

"When you have a dwindling resource, contracting out is a viable alternative," City Administrator Elijah B. Rogers explained. "If we didn't look at an appropriate alternatives people would say we were irresponsible."

Yet, at the same time, Barry, who is trying to broaden his base among black voters in the city, faces serious criticism when he eliminates jobs in an effort to save money. Oppopsition to that principal was one of the major reasons why Barry withdrew his trash proposal.

Trash collection jobs were among the first entry level positions blacks could get in District government, and only recently have they won significant improvements in working conditions, union leaders said. Moreover, unionized jobs are one of the few areas where blacks lacking considerable formal education can make $9 or more an hour.

Barry's proposals to cut costs in the D.C. Department of Environmental Services would originally have eliminated 132 jobs in the trash collection division and 151 more in sewer services.

Barry said he is not fazed by the union leaders rhetoric about the horrors of his proposals. "If the union had a different position, I would be surprised," he said. "What I think I'm going to do in the future is propose massive tax increases and see if people are prepared to pay for all this."

But most union leaders do not think Barry is anti-labor, especially those representing unions that endorsed his candidacy.

One ranking union leader, who asked not to be named, said Barry might find surprising support for his fiscal conservatism among union workers in the city other than those who work for the District government.

"No one is ever completely happy with his boss," the union leaders said. "But you'd be surprised how many labor members who also pay taxes would be in favor of keeping taxes down period."

If the Barry administration does turn to contracting, Rogers said,it would attempt to cushion possible job losses by first trying to find other jobs for all those whose posts are eliminated and urging the contractors to hire among its work force persons being laid off by the city. There is also hope in the administration that many of the contracts will go to minority-run firms.

But Boykin complains that private contractors pay much lower wages, wind up costing the city more. She is digging in for a tough fight.

"It deals with the struggle of black men more than it does anything else," she said typically during one recent interview. She was talking about the possible job losses among trashmen.

"The notion is to contract out to minority businesses. But here you've had some men go through a lot of hell and now you're going to throw that out," Boykin said. "I don't see how that's going to happen without some bloodshed."