I recently planted a Basil plant into potting soil that had been grown hydroponically. It was fine for a week, than in the past two days it started growing a white fuzzy mold on the surface of the soil. See attached pictures. How can I fix this, Please help!

The potting soil you used was not sterlized, leaving bacteria within. I would repot it very gently into a good potting soil. Either that or just scoop off the mould. This will not harm the plant.

Agree it's either fusarium wilt or verticillium wilt. There are fungicides to control this on ornamentals but they are not safe for food crops. Best to start with new pots and trays when you grow things, or dip your old ones in bleach and then rinse them, and use sterilized potting soils, can also bake your regular potting mixes in an oven or pour boiling water over them and let sit covered until it cools off before using it. Basil is supposed to be easy to grow from cuttings, maybe clip your plant and root the stem in a glass of water then replant it fresh?