It was a night to forget for Karius in Kiev, where his two errors helped Madrid to a 3-1 victory and their third successive Champions League title on Saturday.
Karius was tearful post-match as he apologised to the travelling Liverpool fans after he bizarrely threw the ball in off Madrid striker Karim Benzema for the opening goal in the second half.
The German was also at fault having allowed Gareth Bale's second goal – a long-range effort – to escape his grasp, after the Welshman had sensationally restored Madrid's lead following Sadio Mane's equaliser.
Asked about Karius in the aftermath of his second Champions League final loss, manager Klopp told reporters: "I have only very, very few words after the game but it's nothing to talk about.
"I really feel for him, nobody wants that, [but] that's the situation. The mistakes were obvious, we don't have to talk about that, it's all clear; he knows it, I know it, you all know it.
"Now, he has to deal with it, we have to deal with it, we will do that – of course we will be with him, there's no doubt about that. It was not his night, obviously."
Meanwhile, Germany legend Oliver Kahn, a Champions League winner with Bayern in 2001, admitted Karius' performance in the final is the stuff that could ruin a career.
"I'm lost for words," Kahn said, speaking on ZDF. "I can't remember having experienced something more brutal from a goalkeeping point of view than in this final.
"An evening like this can destroy a career."
Liverpool's goalkeeping situation has been problematic throughout the season, after Klopp rotated between Karius and Simon Mignolet.