In my opinion if there are credit unions and small banks in your area, preferably non-publicly traded, they are a far safer bet than the big boys of Wall St. Credit Unions and small banks make their money the old fashioned way via mortgages, and small business and consumer loans. They do not play the market nor do they play those derivative games that few if any understand. They also tend to keep a lot more in capital than is required by the regulators. Non-publicly traded ones don’t have to play the quarterly market games which sometimes sacrifice the future viability of an institution in exchange for bonuses the decision makers will earn today.
Regulators (both Federal and State) audit small banks far harder than they do the big boys, because they can. The slightest infraction gets them written up. The end result is small banks work hard to play by the rules. The big boys consider paying fines a cost of doing business. They make a billion via outright fraud, pay a hundred million as a fine, and keep the rest. Rinse and repeat. They know they are too big to fail, too big to jail.
You can stay abreast of the financial status of your bank by checking these sites. You want a 4 or 5 star ranking. Run fast at a 2 or lower. Note that the ranking are only updated quarterly and even then with a month or two lag in getting the rankings posted, but banks generally do not change their status very quickly.
There are others too like this. These two are very easy to navigate.