Nine people have been charged with various offences and Bernstein has called for an improved relationship between players and supporters.
"We have made good progress but I think there's still room for better behaviour on the pitch," Bernstein told Sky Sports News in a wide-ranging interview at Wembley Stadium.
"Respect is a really important word. Respect between players and referees, players and players and between fans and players. If we could establish that greater level of respect throughout the game a lot of the things we are talking about I think would disappear."
Bernstein revealed that England manager Roy Hodgson will join Stuart Pearce, England's under-21 coach, and Sir Trevor Brooking, the FA's Director of Football Development, for a meeting with Premier League managers next month.
The FA will hold talks about the availability of players for non-competitive internationals including Brazil, the Republic of Ireland and Scotland, in the organisation's 150th anniversary.
"I think it just needs reasonable co-operation," said Bernstein. "Someone like Roy Hodgson, who himself has managed international teams and clubs from all over the world, he understands the issues and problems, so I think it's give and take."
Bernstein has also hinted that The FA may offer former England captain David Beckham a role within the game's governing body.
Beckham has 115 England caps and won his second Major League Soccer title in his final competitive game for LA Galaxy earlier this month.
"It will be very interesting to see where he goes next and what he does next," said Bernstein. "He's an asset English football should treasure. I hope, at some stage, perhaps, The FA and David Beckham can get close together. But that's a little way down the road."
Pushed on further details, Bernstein joked: "No, no (laughs). It's just a hope and it's not based on anything tangible."
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