I entered the information on turbotax. WV did not withhold state tax but it did federal for my unemployment. I did a non-resident return and it calculated state tax (I think). It is now showing a refund in KY and that I owe in WV. Is that correct? I'm a little confused by it.
It depends. Kentucky(KY) has a reciprocal agreement with West Virginia (WV). This means that your wages (not the unemployment) are not subject to tax in WV because they recognize that you will be taxed on your wages in your resident state of KY.
The unemployment is derived from WV sources and that means that it must be included on your WV tax return because it, unlike the wages, does not qualify for the reciprocal agreement.
The WV should be marked as a nonresident and would tax only the unemployment income. Because this is also taxed on your resident state of KY, you will be entitled to a credit for tax paid to another state. KY allows this credit so that you do not pay tax to another state on the same income. The credit will be the lesser of the actual tax liability on the WV return or the amount of tax that KY would have charged for the same income.
When entering the W-2 be sure to enter it accurately based on where the state tax was paid (it will show the state and state withholdings for each state or only one state.
Make sure that WV is completed first, then your resident state KY. If necessary delete your KY return and confirm the WV is taxing only the unemployment compensation.
Then add by the KY and confirm the credit is applied for the tax paid to WV. It is possible for you to have a refund on KY and a balance due for WV since, according to your statement, no WV tax was withheld.