Fathers 4 Justice, the campaign group which fights for fathers' rights to see their children, has distanced itself from the plan to kidnap the Prime Minister's son.
Founder Matt O'Connor condemned the idea of kidnapping unreservedly, and suggested that this could spell the end of the group.
"We do peaceful direct action with a dash of humour," he said. "We're in the business of uniting dads with their kids, not separating them."
Mr O'Connor, 38, set up F4J in 2002 following a difficult divorce which left him without regular access to his two children.
He now feels the organisation, tarred with the extremist brush, will have to consider its future.
The plot to kidnap five-year-old Leo was foiled by police. The plan is alleged to have involved extremist sympathisers of the Fathers 4 Justice campaigning group.
The intention was to snatch Leo, the Blairs' youngest son, and hold him hostage to highlight the plight of fathers denied access to their children. It was understood the men only wanted to hold him for a short period before releasing him unharmed.
Special Branch officers apparently stumbled across the plot while investigating individuals on the more militant fringes of the Fathers 4 Justice group, according to The Sun newspaper.
"There's evidence to support the fact that there was something going on, because SO13 officers [the Metropolitan Police's anti-terrorist unit] had visited some ex-members of the organisation over the Christmas period," said Mr O'Connor. "That had set alarm bells ringing. We contacted Scotland Yard to see if there was anything we could do to help them."
Mr O'Connor said members would now assess if they could carry on with the campaign. "We have to consider the future of the organisation if our name is being associated with such actions. If it gets to this kind of level, then I don't want to be associated with it. We would rather just say, 'We've done the best we can'. This does profound damage to the campaign and is very distressing."
Mr O'Connor said they had been experiencing increasing trouble with a " dark underbelly" in the group. "There are some ex-members who we turfed out last year. We had a bit of a clear-out in May of people who wanted to take the organisation on a more militant route. Other people wanted to go to the dark underbelly of the campaign. It's been a cause for concern for us for some time."
He refused to identify any of the people who were ejected.
Edward Gorecki - who scaled the Royal Courts of Justice dressed as Batman in 2003 to campaign for fathers' rights - said he didn't believe any members of Fathers 4 Justice would have been involved in such a plot.
"I think anybody who kidnaps anyone is disgraceful, and they should be put in prison. Fathers 4 Justice are a non-violent direct action group. They don't seem the kind of guys to do that at all."
Mr Gorecki added that, although he had worked with the organisation before, he was not involved in day-to-day campaigning for it.
Fathers 4 Justice was formed in Britain in 2003, and rapidly achieved publicity through dramatic public protests. In May 2004, condoms full of purple powder were thrown at Mr Blair during Prime Minister's Questions in the Commons. Father's 4 Justice claimed responsibility. In September 2004, Jason Hatch, who was dressed as Batman, managed to climb on to the balcony of Buckingham Palace.
The Blairs have three other children: Euan, 21, Nicholas, 20, and Kathryn, 17. The Met and the Commons refused to comment on the kidnap threat against their brother Leo.