Hardheaded political considerations influenced the White House decision to endorse a bipartisan compromise bill extending and strengthening the 1965 Voting Rights Act.

It appears to be one of the key elements of the Republicans' Southern Strategy for the 1982 elections: try to avoid further offending blacks and whites who support civil rights causes and motivating them to turn out to vote for Democrats.

GOP political operatives inside and outside the White House figure that there are at least 50 congressional districts, a significant proportion of them in the rural South, where the black vote can be pivotal, and that the voting rights issue particularly is one that brings blacks to the polls.

This was clearly demonstrated last summer when the Republicans lost a congressional seat in Mississippi to Democrat Wayne Dowdy.

Blacks turned out in unusually high numbers to vote in a special election that turned out to be something of a referendum on voting rights. Dowdy campaigned in support of a strong bill; his Republican opponent opposed it.

In Virginia a large black turnout helped elect Democrat Charles Robb as governor in another race in which voting rights and race were issues. This was followed by the adverse public reaction to the administration's decision to lift the ban on tax exemptions for segregated schools.

For White House political operatives these were hard-learned lessons on the power of race in politics, even though there are divergent views among Republicans as to how much of the black vote the GOP reasonably can expect to win.

According to one view, there is little chance that Republicans will draw a significant portion of the black vote from the Democrats. But, according to this view, it is important that the GOP not be perceived as anti-black. Such a perception could bring larger numbers of blacks to the polls to vote for Democrats--enough to turn an election.

The wisdom here: "You can't win the black vote, but don't stir it up against you."

For those who hold this view, the technical objections raised earlier by the administration with regard to the voting rights extension had offended blacks and some moderate white Republicans needlessly, and could have hurt the GOP at the polls this fall.

When the Senate Judiciary Committee fashioned a bipartisan compromise on the voting rights issue last Monday, the White House moved quickly to embrace it.

"We didn't want to run the risk of being misinterpreted again," one aide said.

Until the compromise, the White House had argued that the House bill's standard for proving discrimination against blacks or other minority voters was so broad that it could be interpreted by courts to require proportional representation by race in local governments across the country.

President Reagan said Monday that he believed the compromise contained language to prevent that from happening, although some conservatives disagreed.

Some Republican operatives see the possibility of winning at least a larger share of the black vote from the Democrats even if majorities are still out of reach. They point to margins of up to 30 percent of the black vote that liberal and moderate GOP governors such as William G. Milliken of Michigan and Richard L. Thornburgh of Pennsylvania have received.

An indication of how racial issues are regarded in the upcoming elections is the fact that the Republican National Committee has brought a black political strategist, Ron McDuffie, aboard for the November elections.

McDuffie worked in the 1980 presidential campaign and helped in the preparation of Reagan's highly praised speech on black issues to the National Urban League.

His task includes advising GOP congressional candidates with large numbers of blacks in their districts and attempting the feat of getting at least one black Republican elected to Congress.