Despite Liverpool protests, Barkley escaped with a booking for a heavy challenge on the home defender's ankle five minutes before half-time in his side's 3-1 loss at Anfield.
Koeman and Klopp disagreed on the touchline during the match and the Everton boss felt Klopp and Liverpool's bench were too vocal when arguing decisions.
"It is part of football - I also saw some tackles from Lucas," Koeman told Sky Sports when asked about Barkley's challenge.
"That is all about football but we don't make a show like the bench of Liverpool about faults that happen on the pitch.
"We are different to them. We are more into the game and not as into referees, linesmen and tackles. It was football - a hard but fair game.
"Yeah [Klopp and I disagreed] because I don't like coaches from the bench the whole time shouting to referees and linesmen, making a big show.
"It was one tackle and they were crazy. They didn't need the physio on the pitch. It is not what I like. It is a man's sport and you have to have behaviour like that."
Koeman felt Everton's performance was not reflected in the comfortable margin of Liverpool's victory.
"I am proud, really proud about the team," he said. "Of course we lost but the final result is not reflecting our performance.
"They have some quality players, we conceded in my opinion too easy the second and third [goals]. It started with our mistakes in both positions and they punished us for this.
"We made it really difficult for Liverpool. Three at the back worked really well. We did not give them the time to build up, they had to play the long ball - but the position of [Philippe] Coutinho is difficult because he is so clever and plays between the lines.
"It wasn't the Liverpool that we know and that is because Everton did well."