I have a brokerage account with my old address on it from a different state (New York) than my current residence. Therefore the 1099-B and 1099-DIV were issued with the old address on it and I assume the tax information was sent to New York State's tax offices. Since I was not a resident of New York during the year, I do not rightfully owe New York any taxes for these gains. Can I just ignore this and file the taxes in my state of residence or do I have to fill out a New York return and adjust the income to $0?
You can ignore it. If they contact you with questions - you explain that you were not a NY resident and they're not entitled for anything.
They may want to audit your residency status, but there's nothing much you can do to avoid this if they want to do it.