On the sidewalk outside the storefront headquarters of the Rutherford County Republican Party, Ed Cunningham was perplexed.

Cunningham, a dyed-in-the-wool Republican, simply couldn't understand why Rep. Robin L. Beard, the GOP Senate nominee, hoped to get elected by talking about abortion and school prayer.

"I don't give a damn about either one," the retired rubber worker said as he waited for a Beard rally to begin. "As far as I'm concerned they're just campaign rhetoric. Of course, this is the Bible Belt, and some people care about them, but a lot of us think they're of no consequence."

This Senate race has become one of the most vicious in the nation. Beard, finding himself far behind in public opinion polls, has adopted the most controversial tactics and issues of the New Right in a high-risk attempt to overtake incumbent Sen. Jim Sasser (D-Tenn.).

Beard has hired, as a campaign aide, the head of the Religious Roundtable, a New Right group, who has called Sasser "a murderer" and a man with "a perverted conscience who votes in favor of slaughtering little babies." Another Religious Roundtable leader, during a Beard-sponsored tour, accused Sasser of "racial genocide."

Beard, a five-term congressman, has broadcast television ads accusing Sasser of favoring the busing of children for school integration, opposing prayer in public schools and supporting aid for "Communist Cuba" and "Marxist Angola." Each ad distorts, in part, Sasser's record.

"Want to stop busing? Stop Sasser," one ad says. In another, a Fidel Castro look-alike lights a cigar with a dollar bill and thanks "Senor Sasser" for foreign aid.

Sasser, who narrowly upset Republican Bill Brock in 1976, appears genuinely disturbed by the attacks.

"They seek to portray me as ungodly, unpatriotic and in favor of racial genocide," he said late one afternoon in his Senate office. "What worse position could you be in? You're against God, country and little children. I think it's hurting and moving some votes."

The question is whom it's hurting the most. The attacks cost Beard the support of The Nashville Banner, one of the state's most conservative newspapers, and made it difficult for him to establish a strong base among moderate Republicans in eastern Tennessee. In an unusual move, the Banner withdrew its endorsement of Beard, accusing him of "campaigning on outdated conservative demagoguery," and airing commercials "sadly lacking in taste and substance."

Curiously, Beard earlier had rejected offers of help from such New Right groups as the Committee for Survival of a Free Congress.

"He didn't want us. He was afraid of the media attention we'd bring," said Paul Weyrich of that group. "Now he has the worst of all possible worlds: he has all the bad publicity and no help."

Yet Beard plans to continue pursuing New Right social issues. Eddie Mahe, his political consultant, maintains that 6 to 8 percent of the electorate in a state like Tennessee can be swayed by issues such as school prayer and abortion.

"These are . . . issues that help define a candidate," he said. "If a candidate is wrong on these issues it raises fundamental questions about whether he is wrong on other issues and philosophically out of step with the state."

It's unclear whether the tactic has worked or has backfired for Beard. A poll by Keckley Market Research, a Nashville firm, last week reported that Beard had gained 8 points on Sasser since the negative attacks began.

But a more recent poll conducted for Sasser by Peter Hart Associates, the Democratic pollster, indicated that Beard's strength had dropped 6 points during the last two weeks and that Sasser now leads by more than 2 to 1. It also showed that negative feelings about Beard had skyrocketed 15 percent.

Sasser, meanwhile, has used some hardball tactics of his own. He has tried to portray Beard as a hothead extremist with little respect for facts. One Sasser radio ad, for example, states:

"When Robin Beard gets angry he starts calling people names. He once called a secretary of state a 'political tool,' a secretary of the army 'a liar' and a senator from our neighboring state of Kentucky 'vulgar.' Now Robin Beard is calling Sen. Sasser names. Maybe that's because he has so little else to say."

In an interview here, Beard said, "He is just trying to paint this picture of fear. That's all his campaign is about. The real distortion is how Jim Sasser has tried to portray himself. Now people are finding he isn't what his press releases say he is."