San Jose State University has agreed to pay $1.6 million to 13 female student-athletes who alleged that they had been sexually harassed by a former athletic trainer, federal prosecutors and the university said on Tuesday.

In a letter to California’s state university system, the Civil Rights Division of the U.S. Department of Justice concluded that the university had failed for more than a decade to respond adequately to reports of sexual harassment against the trainer and violated Title IX, a law that prohibits sex-based discrimination in federally funded schools.

The university, the letter stated, did this “despite widespread knowledge and repeated reports of the allegations.” As a result, student-athletes experienced “further sexual harassment,” the department said.

Starting in 2009, the Justice Department said in a statement, student-athletes had reported that the trainer repeatedly subjected them to “unwelcome sexual touching” of their breasts, groins, buttocks and pubic areas during treatment in campus training centers.

The investigations by the university and the Justice Department identified 23 student-athletes who they said had been inappropriately touched by Scott Shaw, the trainer, according to the university. The department offered $125,000 to each of them, the university said, and 13 accepted the offer.

Mr. Shaw, who was the university’s director of sports medicine until he retired last year, and his lawyer could not immediately be reached for comment on Tuesday evening.

The Justice Department also found that the university retaliated against two employees in its athletics department, one of whom had repeatedly alerted school officials to the threat posed by Mr. Shaw, and the second had opposed retaliation against the employee who reported the threat. The second employee, the department said, was fired.

“No student should be subjected to sexual harassment at a college or university in our country, especially by an employee who wields a position of power,” Assistant Attorney General Kristen Clarke of the Justice Department’s Civil Rights Division said in the department’s statement.

“With this agreement, San Jose State University will provide relief to survivors and transform its Title IX process to ensure accountability in its athletics program and create a safer campus for all its students.”

The university said in a statement that it had cooperated with the Justice Department’s review and that the findings were similar to a recent inquiry conducted by an external investigator and supervised by the California State University’s systemwide Title IX compliance officer.

That inquiry, which was completed in April, concluded that the 2009 allegations of improper touching during physical therapy sessions were substantiated, as were the more recent allegations raised during the inquiry, the university’s statement said.

“That investigation also concluded that the conduct at issue violated the university’s policies in effect at the time of the conduct,” the statement said. “We thank all the individuals who courageously came forward during the investigations. To the affected student-athletes and their families, we deeply apologize.”

In recent years, the MeToo movement has shined a spotlight on sexual harassment and abuse throughout American society. Universities have faced their own reckoning, as widespread abuses of students have been revealed in other high-profile cases. The payouts have often been costly.

The University of Southern California in March announced that it would pay more than $1.1 billion to the former patients of a campus gynecologist accused of preying sexually on hundreds of patients, marking what university officials called “the end of a painful and ugly chapter in the history of our university.”

The staggering sum — a combination of three sets of settlements with hundreds of alleged victims of the gynecologist, Dr. George Tyndall — set a record for collegiate sex abuse payouts, compensating a generation of young U.S.C. women.

In May 2018, Michigan State University agreed to a $500 million settlement with 332 women and girls who said they were abused by Dr. Lawrence G. Nassar. The university’s president described the settlement as a step “important for the healing process, not only for the survivors, but also for the university community.”

At San Jose State, investigations into Mr. Shaw’s conduct were conducted by the university’s human resources department and campus police in 2009 and 2010. They determined “there was no wrongdoing,” the university said on Tuesday.

“The D.O.J. finding furthers our need for answers to questions about the original 2009-2010 investigation,” the statement said, “and how the university responded to those findings, which is why S.J.S.U. and President (Mary) Papazian launched an external Title IX Procedure Response Investigation. The investigation is currently ongoing.”

Some faculty members were pleased with the settlement announced on Tuesday but also frustrated that it took so long for the university to settle with victims. Nikos J. Mourtos, the president of the university’s chapter of the California Faculty Association, said he did not understand why university administrations continued to fail to act quickly to stop abusers.

“It’s one thing if one student-athlete comes out with an allegation, you can dismiss that,” said Dr. Mourtos, a professor of aerospace engineering. “A second one, you can dismiss that. But we have a series of allegations and you don’t take them seriously? It seems like the university bent towards protecting this person who was committing the abuses rather than the well-being of the athletes.”

The Justice Department’s agreement also requires that the university improve its process for responding to sexual harassment complaints, provide greater resources to the Title IX coordinator, survey athletics employees to better gauge their understanding of the university’s policies and take “concrete steps” to prevent retaliation against those who lodge complaints.

Ms. Clarke thanked the “current and former” students who came forward to share their experiences “and the employees who increasingly advocated for their students.”

“Because of them,” she said, “San Jose State University will adopt major reforms to prevent such an abuse of authority from happening ever again.”