Rich people use "depositor" banks the same way the rest of us use banks; to keep a relatively small store of wealth for monthly expenses and a savings account for a rainy day.
The bulk of a wealthy person's money is in investments. Money sitting in a bank account is not making you more money, and in fact as Kaushik correctly points out, would be losing value to inflation.
Now, all investments have risk; that's why interest exists. If, in some alternate universe, charging interest were illegal across the board, nobody would loan money, because there's nothing to be gained and a lot to lose. You have to make it worth my while for me to want to loan you my money, because sure as shootin' you're going to use my loan to make yourself wealthier.
A wealthy person will choose a set of investments that represent an overall level of risk that he is comfortable with, much like you or I would do the same with our retirement funds. Early in life, we're willing to take a lot of risk, because there's a lot of money to be made and time to recover from any losses. Closer to retirement, we're much more risk-averse, because if the market takes a sudden downturn, we lose a significant portion of our nest egg with little hope of regaining it before we have to start cashing out.
The very wealthy have similar variances in risk, with the significant difference that they are typically already drawing a living from their investments. As such, they already have some risk aversion, but at the same time they need good returns, and so they must pay more attention to this balancing act between risk and return. Managing their investments in effect becomes their new job, once they don't have to work for anyone else anymore. The money does the "real work", and they make the executive decisions about where best to put it.
The tools they use to make these decisions are the same ones we have; they watch market trends to identify stages of the economic cycle that predicate large movements of money to or from "safe havens" like gold and T-debt, they diversify their investments to shield the bulk of their wealth from a sudden localized loss, they hire investment managers to have a second pair of eyes and additional expertise in navigating the market (you or I can do much the same thing by buying shares in managed investment funds, or simply consulting a broker; the difference is that the wealthy get a more personal touch).
So what's the difference between the very wealthy and the rest of us? Well first is simple scale. When a person with a net worth in the hundreds of millions makes a phone call or personal visit to the financial institutions handling their money, there's a lot of money on the line in making sure that person is well looked-after. If we get screwed over at the teller window and decide to close our acocunts, the teller can often give us our entire account balance in cash without batting an eyelid. Our multimillionaire is at the lower end of being singlehandedly able to alter his banks' profit/loss statements by his decisions, and so his bank will fight to keep his business.
Second is the level of control. The very wealthy, the upper 1%, have more or less direct ownership and control over many of the major means of production in this country; the factories, mines, timber farms, software houses, power plants, recording studios, etc that generate things of value, and therefore new wealth. While the average Joe can buy shares in these things through the open market, their investment is typically a drop in the bucket, and their voice in company decisions equally small. Our decision, therefore, is largely to invest or not to invest. The upper 1%, on the other hand, have controlling interests in their investments, often majority holdings that allow them far more control over the businesses they invest in, who's running them and what they do.