Married filing jointly but wife moved to DE from NJ, how to file the state tax?

Yes, if your husband had work in Pennsylvania (PA) while living in Delaware (DE) then a PA return must be filed. See the tax credit below that DE will provide. The PA return should be completed first.

• Spouses that filed a joint federal return and have different residency periods in Delaware can elect to file a joint, separate, or combined separate Delaware return. If you file a joint Delaware return, you are electing to be treated as if you have the same Delaware residency.
• DE Income Tax Instructions

Delaware does not have reciprocity with any state. During the residency period you must report all income earned so review the information above and decide if you want to file jointly.

DE will provide a credit for taxes paid to another state. State Returns - Your resident state requires you to include all worldwide income. Assume both states require income tax returns to be filed: 

1. Report the income on each state return that is from the nonresident state
2. Report it on your resident state and receive credit for taxes paid to another state.

Credit for taxes paid to another state is allowed by a resident state when the same income is being taxed to another state.  Your resident state does not want you to pay tax twice on the same income. The credit that is allowed will be the lesser of:

1. the tax liability actually charged by the nonresident state, OR
2. the tax liability that would have been charged by your resident state

In most cases complete your nonresident state first.