If my employer paid state taxes to the wrong state for the past six months, indicated on my W-2, what can I do to correct it?

Is your employer willing to correct it and make the payment to the correct state and issue you a corrected W-2?  If so, that would be the easiest way.

If not, you will simply file a nonresident tax return for the state that they paid the taxes to and claim $0 income for that state.  This will allow you to claim a refund for that state.  You will then need to file a return for the correct state.  When you file the return for the correct state, you will then likely need to pay taxes to that state since 6 months worth of taxes were not paid to them.