Van der Vaart, capped 109 times by the Netherlands and now working as a pundit for Dutch broadcaster NOS at the tournament, took aim at Van Dijk's turning speed following a late equaliser conceded against Japan.
"I must honestly say that I was a bit shocked by Van Dijk. That turning... It was a bit of a Boeing 747. I hope he will run a little faster during the tournament," Van der Vaart said.
Klopp, who managed Van Dijk for six years at Liverpool and saw him captain the side during his final season at the club, dismissed the criticism on German television on Saturday as the Netherlands beat Sweden 5-1 to all but secure their place in the knockout rounds.
"I don't know if it's worth naming Rafael van der Vaart at all," Klopp said.
"But if he ever says something positive about a player, I'm willing to take him seriously again.
"You have the feeling that he sees something, which then has to be expressed in a flowery way and then he goes against it. But it's not that important."
Van der Vaart's remarks on Van Dijk were not the only controversy to emerge from NOS's coverage of the Japan match. The 43-year-old also faced criticism after suggesting Japan's players "look alike" while analysing the late equaliser, with the comment drawing accusations of racism.
Discussing the goal in the studio, Van der Vaart had been analysing Micky van de Ven's role in the build-up.
"He loses him completely. You can see him moving around. If you are the one marking him, then you are responsible for your man. Look, he's completely unmarked. Van de Ven is nowhere to be seen. He is completely free to make that header," Van der Vaart said, before adding of Japan's players: "They do all look alike of course, perhaps he thought that."
Van der Vaart later issued a statement saying his intention was "never to insult, hurt, or discriminate against anyone."