So the building next door is being remodeled and your living areas are full of construction dust.
Take a look at the outside of the building and see if there are any signs from the SF Building Dept. If the building was built before 1981, there is a possibility the construction dust may contain lead and asbestos. If you see a water truck spraying the construction site, there is a good chance that building contains asbestos, but not always.
Report the problem to your landlord or property manager. Send them a photo if possible.
Then buy a HEPA only air purifier and keep your doors and windows shut as much as possible.
DON’T SWEEP THE DUST – USE A SWIFFER-TYPE MOP. If the dust has “bad stuff” a broom will make the dust go into the air and up your nose. If you must sweep, wet the area or all dust with a spray bottle. Then clean up the mud.
Please give your landlord time to respond to your situation. If things don’t improve, or if you have a reason to believe the dust contains asbestos or lead, then a call to a local tenant-landlord attorney, the SF Rent Control Board or the SF Building Department. If you are SURE asbestos is present, then call the Bay Area Air Quality Management District.