Karius, 25, will spend the next two seasons in Turkey as he attempts to rebuild his career.
Liverpool's signing of Brazil international Alisson curtailed Karius' inconsistent spell in goal but Klopp, who signed him two years ago, believes a fresh start will prove positive.
"We said goodbye [on Friday]. He's really looking forward to it, which is good. It's good for him," the Reds boss said after Saturday's 1-0 win over Brighton and Hove Albion.
"I really think Loris is a fantastic goalkeeper at a very good age. A lot of people spoke to me about our improved defensive things, but as good as Alisson has been in the last three games, he was not here for the games before that, it was Loris in goal and he did a lot of good things.
"Besiktas have quite an experienced team so I think they have a good chance to win the league in Turkey, so that's cool. Now they have a really good goalkeeper, I'm really happy for him and he is happy with the opportunity as well.
"We have Simon Mignolet here, a wonderful, perfect goalkeeper for us now, plus the two kids. That's the situation for us."