The battle between Republicans and labor unions in Ohio, Wisconsin and other states is ostensibly about public workers’ pay, benefits and bargaining rights. What is really at stake, however, isn’t labor’s income. It’s labor’s influence — not just in the American workplace but in American politics.

Critics of unions cast them as exclusive clubs for which the rest of Americans pay the dues. Wisconsin’s GOP governor, Scott Walker, likes to say that unions are the “haves” and everyone else is the “have-nots.” And it’s certainly true that unions aggressively pursue their own interests — sometimes to others’ detriment. When asked in the early 20th century what the American Federation of Labor wanted, the union’s gruff head, Samuel Gompers, famously replied, “More.”

But unions play another role, too — one more like that of civic groups than private associations. Although they want “more” for their members, they also want to make good middle-class jobs the norm. And the most important way they pursue this larger goal isn’t by demanding concessions at the bargaining table, but by operating as a counterweight to the demands of corporations and Wall Street in the corridors of power. That is precisely why opponents of organized labor are seizing upon state fiscal troubles to try to destroy its remaining clout.

Republican politicians haven’t always been anti-union. In 1954, President Dwight Eisenhower declared: “Unions have a secure place in our industrial life. Only a handful of reactionaries harbor the ugly thought of breaking unions and depriving working men and women of the right to join the union of their choice.” His words remind us that America’s leaders once overwhelmingly believed, happily or reluctantly, in the power of organized labor.

When Eisenhower spoke, unions were a critical source of political capital for ordinary Americans who lacked substantial financial capital. Indeed, according to a number of political science studies, the effort by unions to get sympathetic voters to the polls was one of the main forces behind the high rates of voter turnout — regardless of income or education levels — in the decades just after World War II.

Unions also carried the battle beyond the ballot box. Organized labor was on the front lines during the struggle for universal health care and the fight for Medicare for the aged. They were the main champions of organizing rights for workers and of the gradual transformation of Social Security into a strong foundation for a dignified retirement. Unions even lent crucial support to the civil rights movement, leading one congressional champion, Missouri Democrat Richard Bolling, to later observe that “we would never have passed the Civil Rights Act without labor. They had the muscle; the other civil rights groups did not.”

Decades of research have shown that the economic pyramid is flatter in countries where unions are stronger. In economies as different as Canada and Germany, a sturdy union presence has helped reduce income inequality. The reason isn’t just that unions defend their members. They also create changes in social norms, such as pressures for nonunion employers to match union gains. A recent study by the sociologists Bruce Western and Jake Rosenfeld suggests that, including these indirect effects, labor’s decline may account for as much as a third of the rise in American wage inequality since the 1970s.

Unions also push for broad federal policies that reduce gaps in income and wealth. In the United States, they have resisted the rampant deregulation of financial markets and the soaring growth of executive pay. They have also been one of the few organized voices that has consistently pressed back against the string of tax-cut bills for the rich that began in the late 1970s.

All of this makes the decline of unions — which has been far steeper in the United States than in Europe and Canada — a huge political and economic challenge. Private-sector union membership in America has fallen from roughly a third of workers in the middle of the 20th century to less than 7 percent today. Even including the public sector, the share is just over 1 in 10.

Despite these declines, labor continues to be the only large-scale membership organization consistently representing Americans of moderate means on pocketbook matters. While other civic groups — from fraternal organizations to women’s federations — have virtually disappeared, labor is battered but still standing.

Meanwhile, groups representing the affluent have only grown stronger. In the early 1970s, corporations organized on an unprecedented scale to reshape policy and debate. Almost overnight, they expanded their already formidable presence in Washington, forging bonds with wealthy donors promoting business-friendly ideas. This emboldened conservatives while creating powerful conflicts for a Democratic Party increasingly torn between corporate money and its historical and electoral connection to the “little guy.”

The result was a new Washington — one that tilted tax and economic policies away from middle-class Americans. In a little more than a generation, the richest 0.1 percent of taxpayers saw their slice of the national income (including capital gains) increase from less than 3 percent to more than 12 percent. Perhaps the most destructive legacy, however, was the expansion of reckless practices on Wall Street, which Washington alternately ignored and encouraged.

When the financial crisis hit in 2008, unions were a primary voice urging reform. In the face of aggressive lobbying by the health-care and financial industries, labor sunk a huge share of its limited resources into advocacy groups pushing for health-care reform and greater financial regulation.

Today, the recession has eased but has hardly disappeared. Yet the political climate has already swung toward premature austerity, with Republicans demanding cuts in spending that could, according to multiple estimates, cost many hundreds of thousands of jobs. In poll after poll, citizens say that unemployment is the nation’s top problem; that tax cuts for the most well-off should expire; that Wall Street, not public-sector workers, precipitated the economic crisis; and that Medicare and Social Security should be preserved largely as they are.

It may seem ironic that unions are under attack when Washington seems most disconnected from the economic needs of the middle class. But that’s not an irony; it’s a strategy. Critics of unions, such as Gov. Walker, want to cut government and reduce taxes on the wealthy. (Dire budget rhetoric notwithstanding, Walker’s first item of business was to reduce taxes on corporations and the well-off.) And they would much prefer to do it without unions calling them to account.

Unions surely have room to improve. Justifiably worried about external threats, they have too often downplayed their internal weaknesses. They’re quick to point out, correctly, that public-employee pensions aren’t a major cause of state budget woes, and that public-sector workers aren’t overpaid relative to similarly educated private workers. They’ve been much slower to change undesirable features of union contracts, such as policies that overemphasize seniority and protect poorly performing workers. Still, just as we wouldn’t block entrepreneurs from forming corporations just because some firms pollute or engage in financial fraud, we should not radically undermine the rights of Americans in unions just because some hinder effective responses to economic and policy challenges.

In moments of candor, critics of unions reveal that their agenda transcends budget concerns. House Speaker John Boehner, for instance, decried an economic rescue package for states last year as a “payoff to union bosses and liberal special interests.”

But the goal of union opponents is not to exorcise “special interests” from American politics. It is to protect the special interests that represent corporate America and Wall Street from any serious challenge.

Jacob S. Hacker is a professor of political science at Yale University, and Paul Pierson is a professor of political science at the University of California at Berkeley. They are the authors of “Winner-Take-All Politics: How Washington Made the Rich Richer — and Turned Its Back on the Middle Class.”