Among the more lurid allegations was a claim that the Russians held evidence of Trump hiring prostitutes during a visit to Moscow to urinate on a hotel bed which he believed Barack and Michelle Obama had previously slept in.
The file also claimed that the Trump campaign team had multiple contacts with Russian officials during the presidential election race.
The President has repeatedly denied any collusion with Russia.
Sir Richard, who backed Leave in Britain's referendum on European Union membership, said there was no evidence of Russian intervention in the Brexit vote, but that it interfered in the US election.
He said: "I've not seen anything that convinces me at all that the Russians intervened significantly in the Brexit referendum ... I don't think there's any question that they got involved in the US election."
He added: "I think probably on Putin's part, you know, that was a misjudgment."
Trump's former national security adviser Michael Flynn has since pleaded guilty to lying to investigators during an FBI investigation into potential links with Moscow.
In October it was reported that FBI investigators had met with Steele the previous summer.
"CNN has learned that the FBI and the US intelligence community last year took the Steele dossier more seriously than the agencies have publicly acknowledged," the network reported on its website.
The investigation has been criticised by Republican figures, including Trump, who has described it as a "witch hunt".
It has also been suggested that Hillary Clinton's presidential campaign helped to fund the research that led to Steele's dossier.
Marc Elias, a lawyer representing the Clinton campaign and the Democratic National Committee (DNC) used the Washington-based private intelligence firm, Fusion GPS, to conduct the research, the Washington Post reported.
Fusion GPS is then said to have employed Steele, a former MI6 officer, who produced the dossier. The report did not state how much Fusion GPS was paid for the work.
