At the annual general meeting today, directors were told of concerns about the way pro-Palestinian campaigners were able to shut down entire runs this summer.
The Fringe was warned it could no longer claim to be an “open access” festival if there was a repeat in future years.
One top promoter said protesters “screaming at children ‘you’ve got blood on your tickets’” had “pushed the meaning of peaceful” too far.
Fringe chief executive Kath Mainland was also attacked for failing to speak out about protests by the Scottish Palestinian Solidarity Campaign and others.
Police were called to Bristo Square on the first day of shows to handle a “blockade” aimed at ensuring Israeli performers did not appear at Edinburgh University’s Reid Hall.
The Incubator Theatre show was called off a week later after efforts to find an alternative home for the show failed.
The Pola Dance Company, a separate troupe from Ben Gurion University, was also forced to axe its run of La Karina before it had even opened.
Charlie Wood, the director of the venue where more than 150 protesters turned on the first day of the hip-hop opera production, described their impact as “by far the worst” situation he had to deal with in 22 years of involvement with the Fringe.
The London-based promoter said the right of protest did not mean people should be allowed to do whatever they wish outside venues. Mr Wood said that to maintain its “open access” reputation, the Fringe had to ensure that companies went ahead with shows “regardless of what they are saying”.
He said: “The protests were pretty awful. They pushed the meaning of peaceful. People were screaming at children walking past going to another show, saying ‘You’ve got blood on your tickets.’ We couldn’t make the show work in that venue and we tried very hard to find a venue elsewhere, but it just proved impossible to do so.
“We have to be very aware that in an open-access festival, it doesn’t just mean to say we let people do what they want.
“We have to support the fact that anyone should be able to perform even if someone else says they can’t. That’s what ‘open access’ means.
“Next year there will be an issue with Russia or somewhere else, perhaps even the UK.”
Ms Mainland, who has been largely silent on the issue which has dominated the Fringe, faced uncomfortable questions at the Balmoral Hotel AGM.
Andrew Anderson, a Fringe-goer who turned up to raise the issue, said it was “no longer true” that anyone could come to Edinburgh to perform anything they wanted to. He said: “People who organised that demonstration did so with the aim of preventing performances going ahead and they were successful.
“That is a suppression of freedom of speech and suppression of freedom of expression and is entirely contrary to the principles on which I thought the Fringe was based.”
Ms Mainland said she “absolutely” believed in freedom of expression, but added that it also meant there should be a freedom to protest. She said: “It is not our job to support a boycott or not. The job of the Fringe Society is to support the venues and companies that are here. It is up to venues to take their own position on how they want to pull their programmes together.
“Incubator was unable to continue after one performance because of the protest that was disruptive in the wider area. We worked with the venue when they were trying to find another venue to house the show.”