The income your wife earns in Delaware is taxable by Delaware, as well as by her home state of PA. If she files her DE return as Married Filing Separately, then you do not need to file a separate DE return since you have no DE-source income. In the Personal Info section of TT, she should indicate that she had other state income from DE.
Assuming she is a W-2 employee, the income she earns in NJ is not taxable by NJ due to the reciprocal between the two states. However, if her employer mistakenly withheld NJ taxes from her pay, she'd have to file a non-resident NJ return (claiming zero NJ income) in order to obtain a refund. If NJ taxes were withheld from her pay, then she should also indicate in Personal Info that she had other state income from NJ.
ALL her income is taxable by her resident state of PA, regardless of where earned. PA will allow her to claim a credit for the taxes paid to DE, so she won't be double-taxed.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.