Up to 25 protesters managed to enter the reception area of the building in White City, West London, but police and security dragged them away, The Scotsman reports.
Griffin managed to flee the protests by using a side entrance of the building.
Before the programme was recorded, three people were arrested as anti- Fascist protesters against Griffin's appearance on Question Time.
The Metropolitan Police said three people had been arrested on suspicion of causing actual bodily harm, public order offences and assaulting a police officer, bringing the total arrests to six.
The number of officers on the ground increased significantly as the crowd grew in numbers during the evening and a police helicopter circled overhead.
As the debate approached, the chants outside Television Centre grew louder with the crowd yelling: "We're black, white, Asian and Jew, BBC shame on you", and "we won't let Nick Griffin through".
16-year-old Ryan Allen, from Telford, Shropshire, was one of those who managed to get into Television Centre.
"Peter Jones from Dragons' Den was drinking his coffee and he just smiled and looked at us. The police then put us into a corner and dragged us off one by one," he said.
The protests were not restricted to London. About 100 demonstrators gathered outside the corporation's Glasgow HQ at Pacific Quay, while about 30 people targeted the BBC's Edinburgh offices.