Can kleptomania be passed down genetically?

Depends. Depends on the cause. Some neuropsychiatric disorders are inherited some are acquired.
Yes-sort of. Many times children pick up the habits of their parents pof other relatives through modeling. In addition ability to inhibit impulses has some heritability through a number of hypothesized avenues.In addition if a child is taught early on by parents behavior that their needs are unlikely to get met then then later on in life kleptomania becomes one of their strategies to meet thier own needs.