Heather Rabbatts, who was appointed as one of two independent board members late last year, does not however believe there should be fixed sanctions for racist abuse.
Rabbatts, speaking at the Leaders in Football conference in London, said she could not speak about individual cases as the John Terry case is still ongoing - he has until the end of next week to appeal against a four-match ban for racially abusing Anton Ferdinand.
But she added: "Racism still remains a major issue. I think it's probably different in how social media gets used as a force for good and ill.
"We need to be as attentive as to how we deal with it as ever. Not just at the FA but the PFA and how managers are coached to deal with abuse in the dressing room, the role of the Premier League.
"It is a matter for all of football and needs to be top of our agenda. I can't make any comment on individual matters but the point that has been made around zero tolerance is as true today as it ever was."
Former FA chairman Lord Triesman has said Terry's ban should have been longer - Liverpool's Luis Suarez was given an eight-match ban, on the grounds he abused Patrice Evra on several occasions in one match rather than just once.
Jamaica-born Rabbatts insisted, however, there should be no fixed punishment.
She said: No, I don't think so. What we have to ensure is that people appreciate that the tariff structure can stand up to scrutiny to show that people have been effectively sanctioned for abuse."
Meanwhile, Rabbatts backed moves for FA chairman David Bernstein to be allowed to stay on beyond the age limit of 70.
She added: "I think David has shown good leadership, and if he could continue through to the next World Cup, I think that would be helpful in terms of seeing through the governance changes we're trying to put in place.
"Hopefully it will be supported. We'll see what happens. It will be a matter to be debated at council.
"There are a range of discussions about how best to do it constitutionally, but the principle is one I would absolutely support."
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