Soon after sweeping corruption charges were levied against FIFA officials, a top marketing executive for a World Cup sponsor was scolded on Twitter. "You should be doing more than 'repeating concern,'" one person tweeted at Coca-Cola CMO Marcos de Quinto, referring to a statement Coke had issued. "You really have the true power as you are the sponsors!"

Mr. de Quinto pushed back. "Of course," he tweeted. "And we are doing ..."

But doing what? As of Friday, Coke and other sponsors including Visa, Adidas, Budweiser, McDonald's and Hyundai had not publicly divulged specific steps, if any, they were taking against FIFA, whose budget is fueled by sponsorship and TV rights revenue.

The U.S. accuses the sports organization of a vast corruption scheme in which bribes and kickbacks influenced major decisions such as holding the 2010 World Cup in South Africa. After FIFA reelected its embattled president, Sepp Blatter, anyway, Coke issued a new statement: "We urge FIFA to take concrete actions to fully address all of the issues that have been raised, in a swift and transparent manner."

So far, consumers don't seem impressed. The public's response to sponsors has been "'Words are cheap, guys, it's time to do something,'" said Leslie Nuccio, senior content marketing strategist for Meltwater, which monitors news and social-media sentiment.

Recent sports scandals suggest that sponsors could reasonably expect the whole thing to blow over. The next men's World Cup doesn't come for two years, by which point the focus may have moved on. And remember the NFL's imbroglio last fall, when Anheuser-Busch InBev said it was "disappointed and increasingly concerned" about the way the league handled domestic-violence incidents involving players. The NFL and its 32 teams actually grew sponsorship revenue for the 2014-15 season by 7.8% to $1.15 billion, according to sponsorship research firm IEG. TV viewers did not defect.

FIFA hauled in more than $1.6 billion in sponsorship revenue between 2011 and 2014, mostly from World Cup deals, according to IEG. For that spigot to tighten, sponsors would have to decide they were losing more consumer support than they gain by association with the flagship event of the world's most popular sport.

"There's not an easy answer," said IEG Senior VP Jim Andrews. "Soccer fans, like a lot of other sports fans, really do divorce the organization from the sport." Fans might bash FIFA administrators, he said, but "that will not stop them from tuning in the next time their favorite team is playing."

But FIFA's troubles don't stop with the 47-count U.S. federal indictment issued last Wednesday. Activists also allege exploitative, dangerous conditions for migrant workers building infrastructure for the 2022 World Cup in Qatar. The International Trade Union Confederation has estimated 1,200 laborers have died in Qatar since it won its bid to host. Play Fair Qatar argues that sponsors have ethics policies for their own supply chains "but are silent over abuses carried out in their name in Qatar." Others are spreading damning versions of corporate logos on social media, including Coke's white ribbon rendered to look like workers' hands shackled together.

In a statement, Coke said it "does not condone human rights abuses anywhere in the world. We know FIFA is working with Qatari authorities to address questions regarding specific labor and human rights issues." Visa said it has "expressed our grave concern to FIFA."

The FIFA controversies come as consumers, especially millennials, are choosing brands not just for their products but what they stand for. Governance and citizenship account for 29.5% of corporate-reputation scores calculated by reputation firm the Reputation Institute. Marketers probably won't see their reputations hurt unless they are specifically implicated in wrongdoing, said Brad Hecht, VP-chief research officer at the institute. But they have a chance to improve their image by influencing FIFA for the better, he said. And that means going beyond simply issuing statements.