What do President Obama’s White House counsel, President Trump’s national security adviser and campaign chairman, and a finance chairman of the Republican National Committee under George W. Bush have in common with an Iranian-American political scientist who has been a frequent op-ed contributor to the New York Times?

All five of them have been ensnarled by the Foreign Agents Registration Act. That federal law, enacted in 1938 amid anxiety about German influence as World War II loomed, carries criminal penalties. They include prison time and fines for noncompliance.

Mr. Obama’s White House counsel, Gregory Craig, was charged in 2019 with making false statements about work that he and his law firm at the time did for Ukraine and a private Ukrainian individual. A judge dismissed one count, and in 2019 a jury acquitted Mr. Craig of the other one.

Mr. Trump’s national security adviser, Michael Flynn, pleaded guilty in 2017 to making false statements in a Foreign Agents Registration Act filing about his work for Turkey. Mr. Trump pardoned General Flynn in December 2020.

Mr. Trump’s 2016 campaign chairman, Paul Manafort, was charged in 2017 with, among other things, failure to properly register as a foreign agent of Ukraine. Mr. Manafort, too, was pardoned by Mr. Trump in December 2020.

George W. Bush’s RNC finance chairman, Elliott Broidy, pleaded guilty in October 2020 to a count of conspiring to violate the Foreign Agents Registration Act by advancing Chinese and Malaysian interests without properly disclosing the representation. Mr. Trump pardoned Broidy in January 2021.

Now Kaveh Afrasiabi, who has had more than a dozen letters and opinion pieces published in the New York Times, has been arrested and charged with being an unregistered foreign agent of Iran. Mr. Afrasiabi acknowledged to me that he had “received checks” from the Iranian U.N. mission’s bank account, but said he had done nothing illegal and did not believe the Foreign Agents Registration Act applied to his activities. The Department of Justice press release announcing the charges said he had been paid about $265,000 by the Iranian U.N. mission since 2007 and had been covered by its health insurance plan.

The law requires registration by agents of foreign principals if the agents engage in “political activities” for or in the interest of the foreign principal, or if the agents act within the United States as a “public relations counsel, publicity agent, information-service employee or political consultant.”

America has an interest in protecting its political system from manipulation by foreign enemies or their paid agents. Treason and espionage, though, are already illegal, as is bribery. Our political system is gridlocked and unpredictable; no wonder foreign governments or foreign firms are tempted to think hiring the right political consultant or public relations counsel will assure success.

Requiring registration might seem like a moderate compromise between the two extremes of banning foreign-funded influence campaigns altogether or allowing them to proceed entirely unregulated. But that’s misleading. No one is talking about repealing the laws against treason, bribery, or espionage, so “entirely unregulated” is a straw man. And U.S. and American companies conduct public relations efforts overseas, so banning them here entirely would be hypocritical.

American anxiety about foreign influence long predated the passage of the Foreign Agents Registration Act. The existence of the act hasn’t done much to quell that anxiety, let alone extinguish it. For all the concern, though, foreign-paid or foreign-controlled publicity campaigns are minor relative to advocacy efforts that are conducted by Americans for ideological, ethnic, religious, or business reasons.

When so many prominent people run afoul — or are accused of running afoul — of the same law, it raises the questions of whether the law itself is the problem, or whether prosecutors are being overly zealous in enforcing it. Even those tempted to chalk the issue up to a corrupt crowd around Trump are then stuck having to explain somehow the Craig and Afrasiabi cases.

For Congress, there may not be a big political upside in easing rules against foreign political influence in the U.S. All the pressure is in the other direction — tighter restrictions, especially after Russia’s efforts to sway U.S. politics. A bipartisan group of senators has backed an even tougher law.

Yet the existing statute is antiquated, as evidenced by the pre-internet language guarding against the transmitting of “informational materials” by mail, but not by email or social media platforms or the internet. And by targeting political speech, the law itself undercuts the First Amendment.

Unless Congress does rise to review, repeal, or revise the existing law, though, if the pattern holds the statute may eventually wind up entangling some Biden administration figure. A week after the 2020 election, Senator Grassley was asking the Justice Department to evaluate whether Hunter Biden and James Biden should register under the Foreign Agents Registration Act “for work on behalf of the Chinese government or other foreign governments or foreign principals.”

If Mr. Grassley wants to accuse the Bidens of being soft on China, it may yet be a fine political line of attack. But let us debate the foreign policies themselves on the merits, rather than pursuing potential criminal penalties against people for failing to fill out the right paperwork.