Standing on the steps of the state capitol, Jesse L. Jackson today kicked off a voter registration drive that Democratic officials in the region say may be the key to defeating President Reagan Nov. 6.

With Sen. Ernest F. Hollings (D-S.C.), Gov. Richard Riley and state Democratic Party Chairman Bill Youngblood behind him, Jackson called on a small, racially mixed audience to "overcome racial scars, remove the cataracts from our eyes and . . . come together" to elect Walter F. Mondale and his running mate, Geraldine A. Ferraro.

Jackson and state party officials appeared pleased with the start of the voter drive despite the relatively small crowd, about 350 people.

Today's rally grew out of a meeting Jackson had a week ago with southern Democratic Party chairmen. After that meeting, the chairmen and Jackson issued a statement calling for a southern "crusade for economic justice . . . . "

Hollings said only 43 percent of the state's eligible voters voted in 1980, adding that if 1.2 percent more had gone to the polls the Carter-Mondale ticket could have carried the state. Hollings said South Carolina's economic problems and Reagan's portrayal of the government as the "enemy of the people" make it important for Democrats to defeat the president.

"We're proud of our Gerry and the Republicans are proud of their geriatric," Hollings joked at the expense of Reagan, who is 73.

"Our message today is everyone qualified to vote should be registered to vote regardless of his or her station in life," Riley said.

Youngblood said there were more than 300,000 blacks in the state among the 935,000 eligible voters who were unregistered in 1980.

"We've already registered 128,000 new voters this year," Youngblood said, "and we expect the next 30 days could be very good for us." He said Jackson would be a "big plus" in helping to register new voters. "And we find that people who register for the first time vote -- 86 percent in the white community and 85 percent in the black community," he added.

Jackson, repeating an anti-Reagan theme from his presidential primary campaign, urged voters not to be "intimidated by Goliath. He can be beaten."

In 1980, Jackson said, Democrats "did less than our best; the margin of unregistered voters was the difference in that election. We did not vote."

Jackson's trip here was jointly paid for by the national and state Democratic Party, according to Jackson. He said his future registration efforts in the region will be funded by various groups, including labor unions and churches.

After the noontime rally, Jackson met with Steve Patterson, Mississippi Democratic chairman, and Johnny Walls, a black who had been considering an independent candidacy for the U.S. Senate there.

Democrats feared that Walls' entry into the race might take enough black votes away from former governor William F. Winter, the party's nominee, to assure the reelection of Republican Sen. Thad Cochran.

Walls, a Greenville attorney, announced late today that he had decided against running for the Senate.