Not sure what the scope of this discussion is, or how much econ background you have, but modern banks really evolved away from being a secure storage of money a long time ago. What we have now is
http://en.wikipedia.org/w...-reserve_banking fractional-reserve banking, where banks don't ever have enough cash on hand to pay everyone who has an account. Instead they take the money to invest and to give out interest-generating loans. This won't change anytime soon, since we're accustomed to having bank accounts pay interest /to/ us, rather than paying administrative fees to the bank.
If you want something different for your money, perhaps you're looking for credit unions, but they're still a financial institution, although some put more emphasis on creating profit for customers as well as the institution itself.