If you're a part-year resident, you can take the Credit for Income Taxes Paid to Another State in the state where you were a resident while earning double-taxed income.
For example, you lived in Delaware for three months and then moved to New York for the rest of the year. If you didn't earn any New York income while you lived in Delaware, then you wouldn't claim a credit for taxes paid to New York on your Delaware return.
However, if you worked in New York for three weeks early in the year before you moved to New York, then the wages you earned for those three weeks are taxed by both New York and Delaware. You can claim a credit for taxes paid to New York on those three weeks of wages on your Delaware return.