California Gov. Edmund G. (Jerry) Brown Jr. electrified his Senate race against San Diego Mayor Pete Wilson earlier this week with a television ad campaign accusing Wilson of opposing a nuclear arms freeze.

The ad, reminiscent of the one used against Barry Goldwater in the 1964 presidential election, opens with Los Angeles Dodgers third baseman Ron Cey saying, "I want to keep on playing baseball." Conductor Leonard Bernstein says, "I want to go on making music." Actress Candice Bergen says, "I want to go on doing it all."

Then the screen is filled with the fiery mushroom cloud of a nuclear explosion, followed by a scene in which a small boy, surrounded by children, says, "I want to go on living."

The ad closes with a picture of Brown and an announcer intones: "Pete Wilson opposes the nuclear arms freeze. Jerry Brown supports it. Vote for your life. Elect Jerry Brown to the U.S. Senate."

Wilson, who denounced the ad as "character assassination," defended his position by saying, "I want the reduction of nuclear arms to begin. We have to do better than just freeze the nuclear cloud that hangs over the earth."

But Wilson is just one target of the nuclear freeze movement, which has progressed nationwide from scattered protest groups to a potent political force. The Council for a Livable World has raised $320,000 to help 14 Senate candidates who support the immediate nuclear freeze and to unseat several anti-freeze senators.

Its new political action committee, PeacePAC, has targeted 12 congressmen, labeled the "Doomsday Dozen", and solicited money for their defeat in a 500,000-piece mass mailing. The National Committee for an Effective Congress, which contributes to liberals of both parties, also has budgeted $500,000 to defeat its own "Doomsday" list.

One American voter in four will have the opportunity to vote directly on the issue in November, in what will be the closest this country has ever come to a national referendum. It will be on the ballot in eight states -- California, Arizona, Oregon, North Dakota, Montana, Michigan, Rhode Island and New Jersey -- and cities from Philadelphia, Chicago, and Anchorage to Dade County, Fla. and the District of Columbia. The voters of Wisconsin already passed such a referendum in their Sept. 14 primary.

The Council for a Livable World is focusing on Senate races in smaller states where it can have more of an impact. It already has contributed more than $30,000 to Sen. George J. Mitchell (D-Me.) and $30,000 to unseat Sens. Malcolm Wallop (R-Wyo.), Harrison H. Schmitt (R-N.M.) and Lowell P. Weicker Jr. (R-Conn.).

PeacePAC plans a total campaign of $200,000 against its "Doomsday Dozen," who include Reps. William Carney (R-N.Y.), William V. Chappell Jr. (D-Fla.), Don H. (Del) Clausen (R-Calif.), James K. Coyne (R-Pa.), Larry Craig (R-Idaho), John LeBoutellier (R-N.Y.), David D. Marriott (R-Utah), Robert H. Michel (R-Ill.), John H. Rousselot (R-Calif.), Denny Smith (R-Ore.), Samuel S. Stratton (D-N.Y.) and Frank R. Wolf (R-Va.).

However they fare in November, the freeze proponents are here to stay. Their organizing this year, they believe, will give them a political base for the 1984 presidential election. While the movement is nonpartisan, the Democratic Party adopted a pro-freeze resolution at its midterm conference in Philadelphia last June and the two leading Democratic presidential hopefuls, Sen. Edward M. Kennedy (Mass.) and former vice president Walter F. Mondale, have endorsed it.

In Northern California, Republican Clausen is hard-pressed for reelection to the House because the nuclear freeze, which is a ballot referendum in California this fall, is highly popular with many of the affluent, politically-conscious residents who moved to his district from San Francisco and Los Angeles.

Nationwide, many anti-Vietnam war activists and radicals from the '60s and '70s, such as Abbie Hoffman, David Dellinger and the Berrigan brothers, have found a new cause in the nuclear freeze issue.

In upstate New York, however, Democrat Stratton fought off a major effort by nuclear freeze proponents to defeat him for a new term in Congress, whipping his primary opponent last week by about 2 to 1. In his blue-collar working district jobs and the economy are the primary concerns.

Freeze opponents, led by President Reagan, argue that the referenda are meaningless because they have no force of law. Some, including Moral Majority leader Jerry Falwell, are raising money to defeat them.

The House of Representatives voted on two freeze questions last summer. It narrowly defeated the resolution pushed by the movement calling for an immediate "mutual and verifiable" freeze on testing and deployment of nuclear weapons. The House then narrowly approved a resolution backed by the Reagan administration calling instead for Soviet and U.S. reductions of nuclear weapons to equal levels before freezing them at those numbers.

In a recent Washington Post poll of 1,505 Americans, about 29 percent of the respondents said a candidate's position on the nuclear freeze would be the most important influence on their decision whom to vote for, compared to 47 percent who put Social Security on top and 16 percent who put abortion above all other issues.

Other polls, however, show many voters still confused, fearful of the arms race but doubtful that a freeze on nuclear weapons production would work. As the campaign progresses, what was once a motherhood issue few could oppose has become a matter of bitter contention.