Bernstein scheduled a Monday hearing on the issue.
Six people died and 14 were injured June 5 when a four-story brick wall at a demolition site collapsed onto an adjacent Salvation Army thrift shop at 22d and Market Streets.
In the aftermath, the Philadelphia District Attorney's Office has filed criminal charges against the demolition contractor, Griffin Campbell, and a heavy-equipment operator, Sean Benschop; 16 lawsuits have been filed by victims or their families.
Most of the defendants in those civil cases joined Thursday's motion for a protective order, contending public release of discovery material "has already seriously compromised the impartiality of potential jurors in Philadelphia."
"Throughout this litigation, numerous documents produced in pretrial discovery by and among the parties have found their way into the media, often before they were even received by moving counsel," the defense attorneys said in their petition to Bernstein.
They complained specifically about a Dec. 17 Inquirer story based on e-mails provided in the lawsuits' discovery process, in which the property manager for the building owner, STB Investments Corp., described Basciano's impatience with the pace of the demolition project.
The defense lawyers also protested the publication of a video-recorded deposition of Mariya Plekan, who lost more than half her body to amputations after rescuers pulled her from the rubble of the Salvation Army building 13 hours after the collapse.
"Within days of that deposition, the video, transcript, and audio recording of the deposition were released to the press by counsel for Plekan and published by nearly every major print, radio, and television news outlet in the city," the petition stated.
"Plaintiffs' actions have resulted in the exposure of Plekan's videotaped deposition to millions of residents in the area, juxtaposed with sensational editorial commentary."
Lawyer Robert M. Mongeluzzi, who represents eight plaintiffs, said he would oppose the motion.
"This is a public catastrophe that involved the city and public agencies, and we believe that it should be litigated in public," Mongeluzzi said.
Pretrial questioning of potential jurors would be enough "to make sure that you get a jury free of any prejudice one way or the other," he said.