Inquire through the bank directly or with the Federal Deposit Insurance Corporation (FDIC) (see Resources) to see if the bank in question is a member. The FDIC insures the money in your deposit accounts (checking, savings, CDs) up to a certain dollar amount. Banks which are members are required to display the FDIC logo and inform you of current insurance limits. Banks which are not backed by the FDIC can safely be avoided.
Research the banks you're interested in via a trusted online source, such as Bankrate or The Street (see Resources) for consolidated information about their performance over time based on quarterly reports. You can also review their star rating or grade as compared to other banks; a higher number of stars or grade will reflect a better banking institution.
Check the savings and CD interest rates across three or four institutions to see which banks offer the best rates. A lower-ranking bank will often offer higher interest rates to entice you to house your money there while larger, more stable banks offer an average rate because you as an individual aren't providing the bank with the bulk of funds they use for loans.
Review your specific bank's website for further information about its fee structure and interest rates, and search for performance reports which show quarterly earnings over a period of time, such as one year. A decrease in earnings can show a decline in the bank or reflect large numbers of customers making withdrawals to move their money elsewhere.
Know the level of risk you're willing to accept before opening an account or making a deposit. If your research reveals that the bank you're interested in isn't very sound, that doesn't mean the bank will collapse the next day. If you're willing to accept a slight risk (all the while knowing the FDIC is covering your money), then higher interest rates may stand to make you more money than placing your money at a lower-paying, more stable institution.