Air Purifiers do get rid of dust but if you want the Air Purifier to do a good job at getting rid of dust you should use an Air Purifier with a HEPA "High Efficiency Particulate Air" filter to remove the dust from the home.
The HEPA filter will do a good job with your Air Purifier to get rid of the dust in your home as a HEPA filter can remove 99.97% of particles that are airborne.
I use an air purifier with a HEPA filter and I notice my home doesn't get as dusty as it did before I started using the Air Purifier.