Bernstein's comments came barely 24 hours after the FA had charged Manchester United defender Rio Ferdinand over his apparent endorsement of the term "choc-ice" on Twitter. His former England central defensive partner John Terry is also preparing to contest a charge of using insulting language to Ferdinand's brother Anton.
Speaking at Wembley at a gathering of invited guests from FIFA and anti-racism body Kick it Out, aimed at ending all forms of discrimination in the game, Bernstein said: "We must continue to be tough as a regulatory body on sanctioning and disciplining inappropriate behaviour."
He added: "Wherever possible we must bring incidents of discriminatory abuse to charge and all participants must know the consequences of their actions.
"This goes for the Twittersphere as much as on the pitch, in the stands and in the recruitment processes across football."