There are some libertarians, survivalists, and others who advocate avoiding the banking system. It is usually for philosophical and principled reasons. Even some on the left, such as the Occupy Wall Street crowd, would agree with some of the reasoning. There is no question that the major banks are essentially an oligarchy, deriving benefits from the government and the Federal Reserve. The big bankers really do get rich at the expense of the poor and middle class.
It is almost impossible to get by in America today without having a bank account, unless you plan to live in virtual poverty. Almost everything is done electronically today. If you have any kind of a normal job, you are probably highly encouraged to use direct deposit. Even if you get a check on payday, you still have to use a bank to get the money.
There are some who advocate being "underbanked". This means that you just keep a small balance in a checking account and that is it. It means that you keep just enough to pay your bills and regular expenses.
While I sympathize with the notion and I oppose the current banking system being so entrenched with the government, I don't advocate that you make your own life miserable and vulnerable to make a statement. By avoiding the banking system, you are really making things tough on yourself and putting yourself in a position of making yourself poorer.
While I don't think the FDIC should have ever been invented and I don't believe in any kind of government insurance (especially deposit insurance), it is the system we have in place today. It means that bank runs are unlikely, at least in the conventional sense. If there is a bank run, which would really be other financial institutions and companies refusing to loan short-term money, then the FDIC will cover deposits under $250,000 (the current amount since 2008). If the FDIC doesn't have the money, then it will receive money from the government or the Fed. Or perhaps the Fed will simply nationalize the bank.
I think the government would allow a partial default on Social Security checks before it let the banking system fail. That is the number one purpose of the Federal Reserve. If the banking system goes, then the whole system goes.
For this reason, you shouldn't worry too much about having money in a checking or savings account. Your biggest worry is inflation eating away your purchasing power over time. There will not be a Cyprus like event where money is directly confiscated. The Fed does this in a much more subtle fashion.
As long as you stay under the $250,000 threshold at any one bank, then I doubt you have anything to worry about, except for the depreciating value of your money. While it is a personal decision, I don't recommend that you stress yourself out with being "unbanked" or "underbanked". You will probably not make a difference to the system, while just making life hard on yourself. And if you don't use any financial institutions, then you are setting yourself up for trouble by having all of your assets vulnerable to theft, fire, or something else.
I think it is best to diversify your assets. This would include having some cash (digits) in the bank. If the banks have to be bailed out (which they currently are), then you are paying for it regardless of whether you have a bank account or not.