Billionaire Jeffrey Epstein ran a “vast network of underage victims” he paid for sex, using some of them to recruit other juvenile girls to expand his targets, federal prosecutors in New York said Monday.

Mr. Epstein, a high-flying political financier with ties to presidents of both parties, appeared in court, where he pleaded not guilty to the latest charges of sex trafficking and conspiracy.

Prosecutors said dozens of girls as young as 14 were paid to go to Mr. Epstein’s New York and Florida homes, where they were hired to give naked massages — and where Mr. Epstein would escalate the encounter, eventually leading to sexual contact.

He knew the girls were underage, often because they told him so, and he or his employees paid them hundreds of dollars for each encounter and paid if they recruited other girls for him to abuse, U.S. Attorney Geoffrey Berman said in announcing the charges.

“The alleged behavior shocks the conscience,” the prosecutor said. “While the charged conduct is from a number of years ago, it is still profoundly important to many of the alleged victims, now young women. They deserve their day in court. We are proud to be standing up for them by bringing this indictment.”

The charges Monday stemmed from dozens of encounters from 2002 to 2005, but prosecutors pleaded with other victims to come forward.

Mr. Epstein, who served 13 months on sex trafficking charges out of Florida, faces one count of engaging in sex trafficking of minors and a second count of sex trafficking conspiracy. He faces up to 45 years in prison, which Mr. Berman said was likely a life sentence for the 66-year-old financier.

He was arrested over the weekend aboard his private plane at Teterboro Airport in New Jersey. Prosecutors said they will ask a judge to detain him until trial because his wealth and lifestyle make flight a risk.

“When you have two planes and you live much of the year abroad, you know, we think that’s a very real risk,” Mr. Berman said.

The case reaches well beyond legal circles. Mr. Epstein has long-standing ties to major figures in both political parties.

He once counted President Trump and former President Bill Clinton among his friends. He reportedly flew Mr. Clinton dozens of times on his plane, dubbed the “Lolita Express,” from 2001 to 2003. Mr. Trump reportedly flew once and, in a 2002 interview, acknowledged social contact with Mr. Epstein.

“I’ve known Jeff for 15 years. Terrific guy. He’s a lot of fun to be with. It is even said that he likes beautiful women as much as I do, and many of them are on the younger side,” Mr. Trump told New York Magazine at the time, years before his political career.

Neither Mr. Trump nor Mr. Clinton has been publicly accused of complicity in any of the crimes of which Mr. Epstein is accused.

A spokesperson for Mr. Clinton said the former president “knows nothing” about Mr. Epstein’s “terrible crimes” and that the two haven’t spoken in over a decade.

The statement also said Mr. Clinton took only four trips on Mr. Epstein’s private jet and that the Secret Service detail that accompanies all former presidents was with him each time. Fox News reported in 2016, citing official flight logs filed with the Federal Aviation Administration, that Mr. Clinton took 26 trips and Secret Service agents were not listed on some of those logs.

Christine Pelosi, a Democratic strategist, an advocate for sexual assault survivors and daughter of House Speaker Nancy Pelosi, warned her party over the weekend that “it is quite likely that some of our faves are implicated” in Mr. Epstein’s case.

On Capitol Hill, Democrats demanded the resignation of Labor Secretary Alex Acosta, who in 2007, as U.S. attorney in Miami, cut a deal that scrapped a 53-count indictment against Mr. Epstein, allowing him to plead guilty to state charges.

Mr. Epstein served 13 months in a county jail and was required to register as a sex offender. The deal was reportedly reached without the consent of Mr. Epstein’s purported victims, according to the Miami Herald, which published a series of investigative reports on the agreement.

Earlier this year, the Justice Department’s office of professional responsibility said it was investigating whether the prosecutors involved in the deal committed professional misconduct.

Mr. Epstein’s attorney, Reid Weingarten, blasted the charges Monday. He told a federal judge that the plea agreement reached in the Florida case bars further prosecution from the Justice Department.

“To us, this indictment is essentially a do-over,” Mr. Weingarten said at the arraignment. “This is the very stuff that was investigated by the feds in Florida.”

Mr. Berman said he is not bound by any agreement reached by Florida prosecutors, and legal analysts said Mr. Berman is on solid footing because no federal case was ever brought, so there is no double jeopardy.

“The federal government is not bound by state charges or vice versa,” said Steven J. Mulroy, a former federal prosecutor who teaches law at the University of Memphis. “There is a sovereign doctrine that says you can be prosecuted for related crimes at the federal and state level and it isn’t double jeopardy.”

Investigators recovered “hundreds — and perhaps thousands — of sexually suggestive photos” featuring fully or partially nude females, some of whom appear to be underage, according to a court filing Monday.

Some of the photographs were in a locked safe on CDs with handwritten labels such as “Misc nudes 1” and “Girl pics nude,” prosecutors wrote.

They said Mr. Epstein used his wealth to pay incentives to employees and victims for each new girl they brought to him, creating a steady stream of girls to exploit.

Mr. Epstein sometimes scheduled the sexual encounters himself but often directed employees and associates to arrange the dalliances. The indictment labeled three, identified only as Employee-1, Employee-2 and Employee-3. It is not clear whether any of them will face criminal charges.

Prosecutors said they have “detailed, credible, and corroborated” evidence, including messages recovered from Mr. Epstein’s residence with victims’ names and call records.

“The voluminous and credible testimony of individuals who were sexually abused by the defendant as minors, each of whom are backed up by other evidence — will be devastating evidence of guilt at any trial in this case and weighs heavily in favor of detention,” prosecutors wrote.

Although New York prosecutors had to receive approval from the Justice Department to move forward with the case, Attorney General William Barr said he recused himself because he worked for a law firm that represented Mr. Epstein “long ago.” He did not identify the law firm.

The New York U.S. attorney’s office public corruption unit is handling the case, which includes Maurene Comey, the daughter of former FBI Director James B. Comey.

The New York charges drew praise from sexual abuse advocacy groups and an attorney for Mr. Epstein’s alleged victims. They have long argued that he was offered a sweetheart deal by federal prosecutors because of his wealth and political connections.