I work in a state that has state taxes. But my job requires me to work in that state as well as the state next door that does not have state taxes. What is my employers responsibility to track the hours so that I only pay in the state that requires taxes?

You do your own tracking; keep record.

Actually, your employer needs to keep track and allocate your asalary accordingly.

If YOU do it, the state with the income tax will not believe you and require a statement from the employer to back your records.

Work with your employer on this. It is a simple exercise with a spreadsheet to properly allocate where you worked.

By the way, if you live in the state with the income tax, none of the above matters. You have to pay the state income tax on ALL of your income to the state where you live.