Rich people keep SOME money in the bank. Billionaires don't have all those billions in the bank... they're valued in investments and businesses. Millionaires are going to lose their millions real quick if they keep their money in bank savings accounts with shitty rates that barely beat inflation. 

Also, banks only need to maintain the minimum cash reserve in their vaults as dictated by the feds, anything extra is fair game to be invested at the bank's discretion. Though most reasonable banks will maintain a cash reserve at a higher amount than set by the feds, because the requirements can change. Banks usually turn client savings into investments by giving out loans.

Rich people tend to want to get richer, and they invest accordingly. You can't afford the maintenance of that yacht you own if your wealth just declines every year. You can't afford to keep that vacation mansion in Cancun if you're not using your millions to bring in some money. 

Rich people know the value of money, that's why they keep trying to get richer, sometimes even unethically.

A middle-class person is more likely to keep that million in a bank account with shitty rates as "rainy days fund" than anyone else, imo.