Cleaning DURING the remodel will do the most good.

Don't be busting out a vac and trying to dust off the workers and such, just each day after they finish, clean up the dust and debris.

Don't be scared to get in there if when the drywall is removed, you find mold. Bust out the bleach full strength and doctor it up.

Why would you clean before a remodel? Pack up your stuff get it far far away / out of the basement and out of anywhere near any openings to the basement, but there's no reason to have it cleaned.

Do a thorough cleaning afterwards. There will be plenty of dust even if your contractors are meticulous.

Remodeling is messy and you will have contractors coming in and out - no need to clean in advance. I would get everything out of the basement though, and a it's a great time to get rid of things.

I would empty out the space and vacuum. If you will be adding in drywall, the area will get very dusty.

How important is it to do a deep clean before the remodel? It is a partially fi ished sapce with dry wall already with some mold issues that are being taken care of during the remodel.