First, the verification service banks typically use is called ChexSystems. If you’ve been denied a bank account based on negative banking history, you can request a free copy of your ChexSystems report. If there are errors, you can dispute them.
If you have a poor bank history report, try to fix the issues. For example, if you have overdue accounts or collections notices, call the bank listed to find out if you can still pay them back. If they allow you to do this, you can then ask if they’ll remove their negative reports.
You can also find a bank account that won’t check your past banking history, like second-chance banking with Varo.