In 1968, Maryland phone company repairman Harry Beck Jr. became furious when his compulsory dues were used to support the presidential campaign of Hubert H. Humphrey. Beck decided to do something about it.

Today -- 16 years, four elections and 5,000 transcript pages later -- the U.S. Circuit Court of Appeals heard arguments on Beck's case. If upheld, it could result in stronger contraints on parts of the multimillion-dollar lobbying, political activities and organizing efforts of labor unions.

The issue, which may ultimately be resolved by the Supreme Court, is whether a union violates the rights of nonmembers who are required to pay "agency fees" for union representation by using those fees for political and other activities not directly related to collective bargaining.

A blizzard of legal cases has been fought since the 1960s involving union political spending and agency fees, but none of the private-sector cases has progressed as far as Beck's.

"When I found out the union was supporting Humphrey and then-Sen. Joseph D. Tydings, who wanted gun control, I couldn't morally support that," Beck, 43, formerly of La Plata, said yesterday in an interview from his home in Oregon. "It wasn't the money. It was only about a hundred bucks. It was the principle of the union . . . using my dues to cancel out my vote."

Many unions, including Beck's, the Communications Workers of America, already have partial rebate programs that return portions of dues to nonmembers opposed to political spending. But the Beck case may help decide how much spending of nonmembers' money the unions may undertake in the first place, lawyers for both sides said.

The case today before a three-judge panel pitted the AFL-CIO and the 650,000-member CWA against the Virginia-based National Right to Work Legal Defense Foundation, which is representing Beck. The foundation describes itself as dedicated to fighting "the injustices of compulsory unionism," and has participated in dozens of similar cases over nearly two decades.

U.S. District Court Judge James R. Miller Jr. ruled in favor of Beck and 17 coplaintiffs who were also paying agency fees to CWA in March 1983. Miller ordered CWA to rebate 79 percent of the dues the nonmembers had paid over seven years because the union could only prove that 21 percent had been spent directly for collective bargaining with employers. That ruling was appealed. If upheld, Beck's victory could encourage members to drop out of unions because of the financial incentive of paying lowered agency fees, Edwin Vieira Jr., the lawyer for the Right-To-Work group, said.

Laurence Gold, the AFL-CIO general counsel who argued the case today, said in an interview that he doubted that the outcome would cause substantial defections. But he said it could result in financial losses to unions and in the imposition of costly record-keeping systems to segregate political and other forms of spending.

"Their hope and desire is to assure that unions are as weak as possible," Gold said of the Right-to-Work group, "I don't think they will succeed."

CWA has been forced to implement a costly computerized time-accounting system because of the Beck case, and other unions have followed suit. Gold said CWA's accounting system shows that less than 15 percent of dues is spent for political and related causes that may be subject to rebate.

The nation's primary labor law, the 1935 National Labor Relations Act, amended in 1947 by the Taft-Hartley Act, allows unions and private employers to bargain "agency shop" clauses. These require all workers who do not join unions to pay an amount equivalent to union dues in exchange for the union's representation at the bargaining table. Both sides estimate that this represents about 5 percent of workers in agency shops.

But states were given the right to ban such clauses altogether, and 19 states have enacted so-called "Right-to-Work" laws. Virginia has done so, but Maryland has not.

The Supreme Court last April imposed new restrictions on how railroad and airline unions can spend agency fees. Various public employe unions have been limited to using the money on contract-related matters and not for other activities such as organizing and recruiting new members.

But the matter is still unresolved for private-sector unions, which have the bulk of the country's 20 million union members.

The key issue argued today was whether agency fees are a private arrangement between union and employer, or whether it is a governmental action, which would involve constitutional protections against government imposing its will on individuals.