On the campaign trail for mayor of the District in 1978, Arthur Fletcher, Republican candidate, met them again and again.

"People, but especially black people," he said, "they'd come up to me, shake my hand and then they'd say, 'Man, I like what you're saying, but I just can't pull that Republican lever.'"

Few people broke their word. Fletcher lost badly. But he was running with a handicap. The Republican Party in the District is so badly crippled, so weak, that politics in the city is not defined in terms of general elections, Republicans versus Democrats, but as the intramurals of the Democratic primaries. Fletcher is proud that he got 28 percent of the vote in a two- man race, because any Republican candidate has to overcome the fact that registered Democrats outnumber registered Republicans in the city by about 9-to-1.

Fletcher, sitting in the den of his house in Southwest Washington, near a picture of himself with other black Republican appointees in the Nixon administration, says any Republican candidate for mayor has another major burden besides the heavily Democratic voters: the District Republican Pary itself.

"Last time I ran I raised about $60,000," Fletcher says, "Marion Barry raised about $600,000. Let me tell you the reason for that. It's not that Democrats have more money than Republicans. It's that the Republicans are very practical. They look at the city and they see the registration numbers and they realize that any investment of their time and their money--and I mean investment, it's a business deal-- would be a bad investment because it's not likely that their man is going to win. So I couldn't get the money and I couldn't get the professional political managers."

Since Fletcher's unsuccessful campaign, the local GOP has not improved its standing. A deep split over whether the party should support the District's voting rights amendment drive led to a bitter campaign for seats on the central committee. That race alienated some longtime Republicans, and registration has continued to drop, falling from about 26,000 to 21,000.

The party's troubles have served to keep just about all the city's elected officials Democratic.

The down and out state of the Republican Party in the District stands in contradiction to the history of the city. The blacks who poured into Washington after the Civil War were naturally wed to Abraham Lincoln and the party that freed the slaves. But since the days of Franklin D. Roosevelt the Democrats have advertised themselves as the party seeking justice for blacks, aligned themselves with blacks in the labor movement and taken credit for the 1960s civil rights legislation to the point that Republicans here are a rarity.

"In the District," says Marjorie Parker, a lifelong black Republican, "the Republicans in Congress had a big impact in turning the District Democratic with the prejudiced way they treated the city." Parker's father, the Rev. J. L. S. Holloman, was on the city's Republican steering committee, along with other prominent District blacks such as John R. Pinkett, the founder of the real estate business by the same name, and Jesse Mitchell, the founder of the Industrial Bank of Washington.

Today the party has few prominent blacks in its ranks and even fewer on its central committee after the split over the voting rights amendment isolated the remaining blacks in the party on the pro-voting rights slate that was defeated.

"It (voting rights) was not a black- white issue in the party," says Robert Carter, the new chairman of the party and an opponent of voting rights. He says he would prefer that the District seek statehood even if its chances of getting it are slim. "What it really amounts to is not racial but different ways of viewing the future of the Republican Party in the District. . . . I don't think we are going to draw people away from the Democrats by trying to out-liberal them. A lot of people in this town are looking for something other than the Democrats, but we haven't been involved in the kind of things that make people register Republican."

Those things, Carter says, include opposing the inheritance tax that the District government passed and then rejected in a backlash of public opposition. Another job Carter sees for an active Republican Party is taking a stand in complaining about the city's trouble in running elections; attacking the Democrats' inability to get the water bills right; emphasizing the rise in crime and the sale of city property below its assessed value; and what Carter calls the lack of representation the city has in its current delegate to Congress, Walter Fauntroy.

To take those issues to the voters and let them know that the Democrats in office are responsible, Carter says he plans to bring 1,100 college students to the District in January to go door-to- door and register Republicans. His goal is to raise registration from the 11 percent range to about 35 or 40 percent, close enough to give a Republican candidate a realistic chance in next year's races for mayor and congressional delegate.

For all its vigor, Carter's approach to revitalizing the party is not popular with all Republicans. Paul Hays, former chairman of the party, says Carter and the party's new leaders are in danger of "falling on their sword over something like the tutition tax credit. . . . No one in the city was looking for the Republican Party to take a stand in favor of that. It was lucky we didn't."

In Hays' opinion, the future for the Republican Party is in "gradualism"-- gradually building a responsible party that can attract people. To take a local party with a history of being little more than a fund-raising organ for the national party and suddenly plunge it into city affairs will do nothing but encourage suspicion, Hays believes.

But Hays, like Arthur Fletcher, like Robert Carter and just about every other Republican in the city feels there is one bright hope for the future of the party: Vincent Reed.

"He is the one guy," Fletcher says of the former school superintendent, now a Reagan appointee as an assistant secretary of education, "that could make the national party stand up and take notice and put money and the political people in here. He's got a chance to be mayor despite the Democratic dominance. He could win here and by 1984 he would be a national black Republican figure. Black people everywhere would notice him."

Whatever chances a Reed candidacy might have, it would at least help to make the District into a two-party town. That would upgrade the level of politics here--now limited to Democratic infighting--and improve the quality of city government by keeping Democratic politicians on their toes.