The 27-year old was the subject of an investigation from Humberside Police relating to comments made on the social networking site, an investigation that was then handed over to West Yorkshire Police as Cockayne resides in Normanton.
The matter was also the subject of an investigation from the Crown Prosecution Service (CPS) who concluded that no action was required against him.
Following this outcome, the club carried out their own internal investigation and have reprimanded the player accordingly.
Hull KR Chief Executive Mike Smith said: “Ben has been fined the maximum amount possible under club and RFL rules and received a final written warning with regards to his future conduct.
“Ben is also required to attend a diversity awareness course and will spend ten hours a week within the club’s Education and Sport Trust working in the local community.
“The club subscribes to the diversity and equality policy set down by the RFL and is mindful of its responsibilities as an employer and also as a community club.
“We believe the punishment handed out to Ben reflects how seriously we take these responsibilities.
Smith continued: “The club will be reviewing their social media policies with regards to employees and at present has reserved its decision whether to offer Ben terms of a new contract beyond the end of the current season.”
Cockayne commented: “I am sorry and deeply regretful for any offence I have caused with what has happened.
“I have let the club, the fans and my family down. I hope people will understand this was a lapse in judgement and that once again, I am unreserved in my apologies for any offence this has caused.
“I am not a racist person and have many friends of different ethnic origins. They would concur that I am not a racist individual and I hope people that don’t know me can appreciate that.”