Klopp is the best pick. He's a proven manager, knows how to build a team, and has elite level football experince.
Ancellotti wouldn't work imo. He's good at making stars get along and play well together, but he lacks some of the other attributes a Liverpool manager would need.
Mazzarri would probably do well, but I don't think you'd really step up with him. Just the occasional FA Cup and 4th spot. A good choice, but not an ambitious one.
De Boer is promising, but he's too much of a coin toss. Wouldn't risk it given the circumstances Liverpool is in, although he could be a decent choice if you want to give him a long project.
Klinsmann is garbage.
imo, for Liverpool's case: Klopp > Mazzarri > De Boer > Carlo >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Klinsmann