I live in Georgia but work in Florida do I have to pay Georgia state income tax on my income?

Yes, if you are resident of Georgia, you must pay Georgia taxes.

If you are present in a state, then any income earned during this period, must be reported to the state. In your case FL does not have state tax.

To your resident state, you must report your worldwide income for the year. Read: Your U.S. Tax Return: Working or Living in Two or More states

Do I have any responsibility as an employer who has an employee working for me in Florida but lives in Georgia to withhold Georgia State taxes. [email protected]

According to the State of Geoirgia - yes, you should. From:
https://www.google.com/url?sa=t&rct=..._FdA4xs7YODMWw

Georgia residents are subject to the withholding tax laws of the state in which they work. If that state does
not require withholding, tax should be withheld and paid to Georgia. O.C.G.A. § 48-7-100(10)(H)