The Newport East Labour AM is a fierce critic of the BBC decision last year to have BNP leader Nick Griffin on the Question Time panel.
He believes the party must not be presented as part of the political mainstream.
He challenged rival parties and organisations hosting hustings to deny the BNP publicity.
Both the Conservatives and Liberal Democrats said they would debate with the BNP with the aim of exposing it as a party of “thugs and fascists”.
Mr Griffiths said: “I certainly call on them not to share a platform with the BNP because I think it does give the wrong message of the acceptability of the organisation.”
The BNP said in a written statement: “[This] just shows how frightened Labour is of voters hearing what the British National Party has to say.”
But Mr Griffiths insisted that the BNP should not be seen as a “bona fide political party”.
He said: “They have had some success presenting themselves as a mainstream party, which they are not. They are getting more publicity through appearances like Question Time.
“I think it’s all very worrying, especially perhaps when we live in an age which is anti-politics and the mainstream parties are viewed with considerable suspicion because of issues like MPs’ expenses and sleaze and so on.”
The Conservatives are prepared to take part in hustings if all the other parties are represented.
Tory Assembly leader Nick Bourne said: “We think their agenda is vicious and needs to be exposed. It isn’t what people in Britain have stood for – I’m absolutely clear on that.
“I think they are nasty, vicious thugs.”
Mr Bourne said views between mainstream politicians differed on how to tackle the BNP but the parties agreed “they need tackling”.
He said: “I think there is a great danger in pretending they are not there. That’s probably what happened in Hitler’s Germany – people pretended they would go way and they didn’t.”
Mr Griffiths was not impressed by recent changes to the BNP constitution to admit people from ethnic minorities.
He said: “I think it’s part of their attempts to mislead people, to change their image and make them think they are something they are not.”
Plaid Cymru does not want their candidates to take part in hustings with the BNP but will not stop them from doing so.
A spokeswoman said: “Plaid Cymru believes that, wherever possible, the party's elected members and candidates should do all in their power to reach agreement with other mainstream parties not to share a platform with the BNP.
“Where this is not possible, and the choice is between sharing a platform and leaving an empty chair, elected members and candidates are free to challenge and defeat the racist lies peddled by the BNP by choosing to appear on a platform with their representatives.
“However, no candidate or elected member would be compelled to share a platform with the BNP.”
The Liberal Democrats were ready to engage the BNP in open debate.
As spokesman said: “If the BNP is invited to an event, the Welsh Liberal Democrats should be there to take them on. The BNP has no interest in the important matters that people really care about – jobs, the NHS, good quality schools and reducing the tax burden for hard-working people.
“We should be there to show people that the BNP are a party of thugs and fascists with no answers to the important issues facing Wales today.”
Mr Griffiths said there were ways to attack the BNP other than taking part in the same hustings.
Wales Online
