MEDINA, Ohio—Karen Dombi was thrilled when her three oldest children were picked for student government this year—not because she envisioned careers in politics, but because it was one of the few programs at their public high school that didn't charge kids to participate.

Budget shortfalls have prompted Medina Senior High to impose fees on students who enroll in many academic classes and extracurricular activities. The Dombis had to pay to register their children for basic...