Karius was at fault for two of Madrid's three goals as they won the trophy for a third year in a row, first throwing the ball straight at Karim Benzema and then fumbling a long-range Gareth Bale shot into his own net.
Klopp acknowledged that Karius was to blame for the errors, but insists that the club will support him as he tries to bounce back from the disappointment.
"What can I say? Loris knows it, everybody knows it. It's a shame in a game like this and after a season like this. I really feel for him; he's a fantastic boy," Klopp said.
"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."
Karius accepted the blame for the mistakes after the match, with Madrid running out 3-1 winners in Kiev.