In Texas’ 28th Congressional District, which stretches from Laredo and the southern U.S. border to the eastern suburbs of San Antonio, tensions are brewing as Representative Henry Cuellar, a centrist Democrat, faces federal bribery charges.

So far, Democratic elected officials, members of Congress and party leaders at home and in Washington have refrained from calling for his resignation. Donald J. Trump has defended him, and even one of Mr. Cuellar’s potential Republican opponents has said Mr. Cuellar is innocent until proven guilty. Many voters don’t believe he will lose his re-election bid.

Yet some local Democrats worry that his case could have consequences up and down the ballot in November. They fear his legal troubles could dampen Democratic turnout in the historically blue, majority-Hispanic counties that have seen a surprising rightward shift in recent years.

Voter apathy and institutional distrust run deep in South Texas. The list of indicted area elected officials, past and present, is long: One study by researchers at the University of Illinois at Chicago ranked the South Texas region 11th in the nation in public-corruption convictions — there were 873 cases from 1976 to 2021.

Mr. Cuellar’s indictment may reinforce the distrust of public officials, local Democrats said, at a time when the party is seeking to engage the larger Hispanic electorate nationwide.

Sylvia Bruní, the chairwoman of the Webb County Democratic Party in Laredo, Mr. Cuellar’s hometown, put it this way: “The most common refrain we hear from people who have not registered or have not voted is ‘De que sirve? Todos están comprados.’” Translation: “What’s the use of voting? Politicians — they’re all bought.”