Clarke was speaking after a Sky Sports News programme showed Millwall fans persistently subjecting Leeds striker El-Hadji Diouf to racial abuse at the Den on November 18.
Footage appeared to show the Senegal international alerting the referee, Mark Halsey, and making stewards aware of his allegations but Millwall insisted no complaint of racist abuse was made by the player.
Clarke, who was at the match in question, told Sky Sports News: “To have our players subjected to such vile and filthy abuse deserves nothing more than the culprits being arrested and facing the full weight of the law.
“There’s two issues, one is we have to continue to work with the clubs on our anti-racism programmes which are agreed with the FA.
“Secondly is the reticence of black players to come forward and feel comfortable in supporting it.
“If we’ve created an environment that doesn’t support black players in confronting this issue we`ve done something terribly wrong.
“We’ve got wrap ourselves around our black players and give them our support to report such episodes and then deal with such episodes extremely severely.”
Clarke has met Kick It Out chairman Lord Herman Ouseley and Professional Footballers’ Association chief executive Gordon Taylor for talks over players who suffer abuse but do not make an official complaint.
He added: “We need to create a culture in football that rather than making the person who points out the problem the villain, they get the support they deserve in being able to point it out. We need to support the victim.”
Clarke said he was aware of lots of shouting and chanting directed at Diouf during the match but did not hear any co-ordinated racist chanting.
“That doesn’t mean there was no individual racist abuse,” he added. “We have to use every tool in our armour, be it CCTV, steward and player statements, to hunt down every racist at every football ground and make sure they never get in again.”
Clarke added that Millwall had worked very hard to tackle a past reputation for a racist element among their fans.
He said: “The club has worked very hard, and spent a lot of money, reaching out to people to improve them. Persecuting the club rather than the perpetrators is a wrong answer.”
Millwall released a statement last night in which they said they were “appalled” by the footage they had witnessed.
Allegations of abuse against Diouf initially surfaced online directly after the match, with Millwall claiming an investigation carried out at the time resulted in there been no case to answer to.
Millwall chief executive Andy Ambler said: “That day against Leeds obviously there was an allegation of abuse made online initially. We investigated it straight away.
“We interviewed stewards, police, players, we looked at the TV footage too as it was live on Sky.
“After speaking to the player who had the alleged abuse at him during the day, he said that he heard no abuse on the day so that was basically at that point the end of the investigation - although we’re going to open it again now.”
Ambler added: “Clearly that behaviour is unacceptable in any football ground in the country and at Millwall we will ban the individuals for life and we hope that Sky will pass the footage to the police so that further action can be taken.”
Meanwhile, the chief superintendent of the Metropolitan Police’s Public Order branch, Mick Johnson, says his force is working closely with football clubs at tackling racism.
Bolton forward Marvin Sordell has previously made allegations of racial abuse towards Millwall’s fans.
Sordell who represented Great Britain at the London 2012 Olympics, has faced racial abuse on social media websites and claims he is not shocked that such things still occur.
He said: “We’re a developed, multi-cultural society. It’s surprising it can still go on - but it doesn’t exactly shock me.
“We can’t be silent about it. We need to make people aware that there is a problem going on and only when people are aware of the problem can it be sorted out.”
Football Association chairman David Bernstein said he was treating allegations with the utmost importance, telling Sky Sports News: “These issues bother me intensely. They have become top of my agenda in terms of moving these things on and combating these things.
“The awareness of it could not be higher.”