Kim, who played in their 3-1 win over Bahrain, tore the anterior cruciate ligament in his knee during a training session ahead of their second game against Jordan on Saturday.
"We're very sad about Seung-Gyu's injury but it's part of sport, it happens in tournaments but we have to move on. We keep him in our thoughts and we fight for him," Klinsmann told reporters.
"What I told the team was that we are here and we have a mission, that is to stay till the end of the tournament."
South Korea midfielder Lee Jae-Sung said the injury to one of their most senior players had motivated them to go deeper in the tournament.
"Being an athlete I know how much being injured hurts. But there are 25 other players in the squad... We're all together in this and hopefully, it gives us extra motivation to go far," he said.
South Korea are in disciplinary trouble after five players were booked in their group opener, including skipper Son Heung-Min and centre-back Kim Min-Jae.
One more yellow card will result in a suspension for the following game but Klinsmann said he would not ask his players to exercise caution.
"We obviously didn't want so many yellow cards. Obviously, if you get a second one they miss a game. It's part of football, two or three could have been avoided - the players know that too - but it's something you have to live with," he said.
"Managing yellows is something for the players to look at. As a player you need to trust your instincts, you can't be too cautious. If it happens, it happens."
Jordan are top of Group E after they beat Malaysia 4-0 and although many expect the Middle Eastern side to play defensively against a team 64 rungs above them in the world rankings, Klinsmann thinks otherwise.
"Jordan will make it difficult and frustrating for us and we need to find a way to break them down," the German said.
"But if you score four goals, you're not a defensive-minded team... If the coach fields a defensive-minded team, that's his choice.
"They've proven they're dangerous, they can counter-attack really fast and have gifted players up front. We have to be alert."
Klinsmann also confirmed Wolverhampton Wanderers forward Hwang Hee-chan is pain-free and back in training after missing the first game due to injury.
