PCI doesn't have anything to do with banking information, and may not be applicable depending on your hosting situation. PA-DSS deals with web applications.
I would very seriously consider the risks involved in this, compared to the monetary benefit you derive as a result of accepting and storing this information. A data breach of this information may expose you to a great deal of risk and expense. For example over 30 states have their own beach notification laws, and timelines for how long you have to contact the state attorney general, customers, etc. You may have only a day or two in many cases and the legal fees alone would be very expensive.
Beyond that, you are often responsible to notify all affected customers in most of these states by mail. Many companies end up purchasing/providing free credit monitoring and you may be sued by banks or individuals whose accounts are compromised in order to cover actual losses, including actual funds and bank resources. You may face additional, personal liability if you don't follow industry best practice, which typically would include code and pen test review of all releases and updates, annual audits, encryption, file integrity monitoring, web application firewall usage, SIEM, IDS/IPS, and the list goes on.
This is not all to scare you, but to convey the seriousness of the undertaking. If you are determined to store banking-related PII I would strongly consider consulting a lawyer or an experienced banking industry IT security professional. At the very least, I would talk to a few of the larger banks in your area and ask what they would require if you were operating a service like this for one of their business clients.
Disclaimer: I'm not a lawyer, views and thoughts are mine and mine only, etc, etc.