to handle money is actually take it out of the register/safe and count it, account for it, deposit it in the bank, etc. To have direct contact with paper and silver in a business or organization.

Вполне можно сказать "кассир занимается деньгами". Я сказал бы cash, но в гугле много примеров и для money.

Другие примеры
Who handles the money in your home? What kind of debt load do you carry? How often do you argue about spending money?

How to Handle Money

Здесь "заниматься" тоже подходит.