If I live in Florida but job is based in Utah do I have to pay state income tax to Utah

Yes, if you are physically working in Utah, even if you live in FL, you will need to file a nonresident return for Utah. 

Thank you for your response. I physically (remotely) work in Florida. The company is just based in Utah. It seems strange that we would pay taxes to a state we don't live in.

If you are not physically working in Utah then no.  It is based on your physical location.