Alamo is suspected of violating the Mann Act, which prohibits taking children across state lines for illegal purposes. Frazier described those purposes as "sexual activity."
He said he didn't believe any children were with Alamo at the time of his arrest but would give few other details.
Federal agents and Arkansas state police had raided the headquarters of Tony Alamo Christian Ministries in tiny Fouke on Saturday and removed six girls ages 10 to 17. They sought evidence that children there had been molested or filmed having sex.
Prosecutors sought Alamo's arrest after interviewing the girls this week, but Frazier would not disclose what the children said.
The Southern Poverty Law Center, which tracks hate groups, describes the ministry as a cult. Alamo's church rails against homosexuals, Roman Catholics and the government, and Alamo has preached that girls are fit for marriage once they are sexually mature.
"Consent is puberty," he said in an interview with The Associated Press last week while agents raided the compound. He denied any involvement with pornography.
An Arkansas judge has hearings set for Friday and Monday on whether the state Department of Human Services can keep custody of the six girls. The girls will attend the hearings.
"We will transport them to and from hearings. We will take part in any future hearings," agency spokeswoman Julie Munsell said. "Our job right now is to basically take care of them."
State Circuit Judge Jim Hudson said two hearings would be conducted Friday and the other four Monday in Texarkana.
The six hearings will be split among three judges who will decide whether the state had enough evidence to temporarily remove the children from their homes on the Fouke compound. If a judge rules against the state, the girls would be returned to the parents.
Arkansas State Police spokesman Bill Sadler said tha no further arrests were planned that would involve his agency.
FBI agents and police in Arizona arrested Alamo as he was leaving the Little America Hotel, which is along Interstate 40, Frazier said. It wasn't known where Alamo was headed when he was picked up.
The hotel, in Arizona's northern mountains near the Grand Canyon, bills itself as a luxury resort. Fred Reese, a hotel spokesman, declined to comment.
Alamo and his late wife Susan were street preachers in Los Angeles before forming a commune near Saugus, Calif. Susan Alamo died of cancer in 1982; Alamo claimed she would be resurrected and kept her body on display for six months while followers prayed.
Alamo was convicted of tax-related charges in 1994 and served four years in prison after the IRS said he owed the government $7.9 million. Prosecutors in that case argued that Alamo was a flight risk and a polygamist who preyed on married women and girls in his congregation.
Since establishing his ministries in Arkansas, Alamo has been a controversial and flamboyant figure in the state. Snapshots often show him wearing large dark sunglasses, and he recently said he is legally blind.
In his autobiography, "My Life," former President Bill Clinton, an Arkansas native, described Alamo as ""Roy Orbison on speed."
Clinton recalled traveling in 1975 to see Dolly Parton sing at Alamo's compound in the town of Alma. Remembering the fiasco after Susan Alamo's death, Clinton wrote: "A couple of years later, he got involved with a younger woman. Lo and behold, God spoke to him again and told him Susan wasn't coming back after all, so he took her out of the glass box and buried her."
FBI documents identified Alamo by his birth name, Bernie Lazar Hoffman, and said he turned 74 the day of the raid. Alamo has said he was born Jewish but converted to Christianity.