Im not familiar with forced hot water heat. So there is an actual baseboard heater along the wall. and not just a vent in the floor?
After I renovated my place the dust that came out of my forced air ducts was pretty intense! drywall dust, sawdust, grout/tile dust. Forced air seems to blow everything around the house
I found one type of filter here:
DustChek Vent Liner Filtration Kit | National Allergy
Ive never used anything like this but I have heard about them. Its worth a try if your having health problems related to the dust and it is a low cost option. Worth a try before you shell out big bucks for an air purifier