Most people would not be affected tremendously if there were no federal deposit insurance. These days, people do not tend to keep that much money in banks.
Theoretically, the lack of deposit insurance should cause people to keep less money in banks. They will want to avoid banks because, if the bank goes broke, they could lose all of their money. Therefore, they would do other things with their money. They would likely keep some of it in money market accounts which allow people to write checks and are therefore rather liquid. They would keep more of their long-term investments in things like stocks and bonds, or maybe in things like real estate or gold.
In reality, though, people do not keep much money in banks anymore. The interest rates on saving accounts are so low that it does not make sense to keep any serious money in banks. People do keep enough money in the bank to pay monthly bills and things like that, but it would be hard to do anything else with that money. People would still need to keep enough in the bank to be able to use their debit cards. They might, however, start to try to find ways around banks in cases like these. They might turn more to things like prepaid debit cards.
Overall, there would not be a huge change in behavior, but money would generally flow out of banks and into other types of investments or other payment options.