The political strength of the District of Columbia's organized labor movement may be fast becoming the first political dinosaur of home rule.

It was a laughing matter for Mayor Marion Barry recently. Barry noted that in this month's special election, strong labor support plus an active labor campaign failed to win the atlarge City Council race for Douglas E. Moore, who had been endorsed by most of the city's labor leaders.

"It's obvious," Barry said, with a rare and hearty chuckle, "that the 'followship' doesn't follow the leadership."

Part of the inside joke was that Barry's energetic political organization, which virtually ran the campaign of winner John Ray, sought especially to win in Ward 7. Many of the city's labor leaders live in that ward, located in the Far Northeast and Southeast sections of the city, and Barry's scrappy political irregulars wanted to send them a message-"Look out labor, here we come."

The special elections marked the second successive time that big labor in the District had gone down to serious political defeat. In last year's intense Democratic primary, labor endorsed then-mayor Walter E. Washington for reelection and threw an estimated $100,000 into the Washington campaign.

But labor's effort to stake out a place in the city's developing political power structure failed, and that failure was footnoted by Moore's defeat this spring.

Labor leaders admit the obvious: "I don't think our game is as together as it ought to be in terms of politics," said William Lucy, international secretary-treasurer of the American Federation of State, County and Municipal Employees, one of the most active labor leaders in District politics.

"Labor hasn't been 100 percent together," conceded Henry Brock, president of Building Laborers Local 74 and president of the political arm of the 200-member Greater Washington Central Labor Council.

The hardest and most telling assessment came from one politically active laborite, who in the candor of anonymity said, "Labor has simply got to come into the 20th century politically. We can only go so many times to the well and come back poorly before we lose our clout."

The present political weakness of organized labor in the District is a story of outdated political style, ineffective internal organization, shifting populations and an individual-oriented group unable to convince its collective membership to vote and act in its own ostensible self-interest.

National observers say that organized labor in the District is plagued by many of the same problems that beset labor on a national level, including an image of leadership that is sometimes perceived as unimaginative and uninspiring.

"It isn't that the leadership is ineffective or their judgment is bad," said one observer. "But the way they deliver the message is undoubtedly not as effective as it might be."

On paper, the potential political strength of organized labor in the District is significant. A poll taken by The Washington Post last June indicated that one of every four regular voters in the District's dominant Democratic Party is in a union household.

Despite its paper strength and its lines to national union political organizers, big labor in the District has not learned to play the sophisticated politics of identifying and working voters well in advance of election day that has become a major style of winning elections here.

"Too much is relied on in terms of hoping that the membership will simply vote for it on the basis of the endorsement of the leadership," Lucy said.

Another problem is the lack of internal organization. Union mailing lists are outdated and inaccurate, Brock said. There are dozens of individual and jealously guarded fiefdoms under the umbrella of the Central Labor Council, others note. Last year, when the Washington campaign needed an up-to-date list of shop stewards from the Central Labor Council, it was barely able to obtain one, insiders recall.

"Labor just isn't organized politically to work together to deliver a message," said one politically active laborite. "They've got a problem of fashioning themselves into a political instrument."

Lucy said some of labor's political problems may have sociological and economic roots.Being a member of a labor union is not always the overriding concern among labor families, he said. To some local union members, for example, administration of the city's Department of Human Resources may be more important than a candidate's stance on the minimum wage.

"Most of our folks are well beyond the minimum wage already," Lucy said.

Union programs to encourage members to know and vote for their interests are considered by some politically astute members to be little more than a sham.

The changing faces of the District's population also has taken a toll on organized labor's political clout. Brock estimated that as much as 90 percent of the membership of some unions live in the suburbs, and many of them have moved there only recently as affordable, low- and moderate-income housing has become more scarce in the city.

"A lot of the people we had in '74, don't live here in '79," said Brock.

Some union leaders are strangely optimistic that one other change in the nature of the city's workers will lead to the political salvation of labor. The increase in union membership, they say, is no longer among the old-line buildings trades but rather among service workers-retail clerks, barbers, meatcutters and hotel workers, for example. That could bring a new, younger and more politically sophisticated leadership into the ranks of the District's labor community before the process of political extinction goes beyond the point of no return.