The problem with online is that there are no consequences. Dominating others is hardwired into our reptile brain to feel good, that is why it feels so good to win a game for instance.
In the real world people don't generelly act like in GTA. Why? Well, first of all we only have one life. If we go around shooting others we know that someone is going to come around and shoot us. In the game, a death is just slightly annoying so self preservation is not a motivation to behave well.
In real life we also have law enforcement. Sure, that is in GTA as well but they just kill you and leave it at that. You don't risk huge fines, losing property or spending years behind bars. If there was a system that placed your character in jail for a week after certain offences I'm sure the behaviour online would be drastically improved. It doesn't even have to be technically complicated: if the cops catch you after you've killed a player in free roam you go to jail for a week (they could even put you in a cell at Bolingbroke and have you play an inmate until your time is served). Special "no rules" lobbies could be created for those who want a constant warzone or an occasional gang war between crews. Around the map, limited free fight areas could spawn as well, so that everyone who wants to fight can go there.
As it stands now, public sessions are not much more than domestic terrorism simulators, which is bad in so many ways.