No, you do not need to file a tax return (or pay income taxes) in that other state, if your only reason is that you received "passive" income from a bank or financial institution that is based there.
The only thing that would change this conclusion is if what you meant to you say when you write "I earned income from a bank that is not my state of residence" is that you actually physically worked at a bank in another state (as in held a job there), and received a W-2 or a 1099-MISC for your effort. In that case you would be considered as having "earned" income from that other state, and in this instance, yes, you would need to file at tax return and pay income taxes to that other state.
The distinction is key here between "passive" income and "earned" income, you see. Passive income of an investment portfolio nature is not subject to being taxed by a state that is not your home state of residence; whereas earned income is.
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