A broom and dust pan is best on hardwood, use a vacuum on carpet and upholstery. Use a microfiber towel on furniture. Whatever you do DON'T use water or any liquid to clean up, it will make a mess. Depending on the state(or country) you live the contractors might have to use a containment enclosure with an exhaust fan if there's asbestos or mold, if the drywall was made before 1978 then it definitely has asbestos(it was still used until the mid 80's).

Make sure you wear a nuisance mask when cleaning, it will protect you from asbestos and mold.

It could still have asbestos, they used it in a lot of building materials:

EPA Bans Asbestos

It also depends what country you live in.

When I was having dry wall ripped out and replaced all over the place!...I vacuumed with a tank vac every day and changed the bag. My contractor hung plastic in strategic spots and covered lots of carpet with plastic, and I laid sheets over every thig else. It was winter andw e couldn't go without the heat....so we changed filters very often....and kept doors to rooms closed. You think ti will never be clean.....but it will!

Dont use a regular house vac to clean up- it will ruin it since it pulls through the motor. A shop vac is fine. Cover everything you can. Drywall dust is so fine it travels everywhere and gets on everything-including vertical surfaces.

Our contractor did all the cleanup. A tiny bit of dust was on the blinds. Have you asked what cleaning they will do?