Deutsche Bank AG played a key role in Jeffrey Epstein’s financial dealings in recent years, helping the accused sex-trafficker move millions of dollars in cash and securities through dozens of private-banking accounts with the German bank, according to people familiar with the matter.

Mr. Epstein, a wealthy American financier facing two counts in New York related to alleged sex trafficking of minors, used Deutsche Bank’s private-wealth division to hold and manage assets connected to him, clients or entities he was affiliated...