Asbestos is only bad if it gets airborne and then inhaled. If asbestos is protected and left in place, there is nothing inherently wrong with it; there are many older installations of asbestos wrapped pipes in industry where the acceptable practice is to encapsulate the asbestos (cover it, wrap it, etc.) with new materials and leave it in place. If you are concerned that your pipe wrap contains asbestos, get a sample of it tested by a licensed lab. If you ever do some serious remodeling, ductwork replacement, etc., then you should have an asbestos abatement company come in to take care of the issue properly and safely. Your pics show what looks like metal ductwork. Surely, the metal ducts are not cracking???? Might they mean the joints between ducts are coming loose and those are the cracks they speak of? And, those ducts do not look to be insulated, which, IMO, they should be since they are in an attic. I wonder if a single, simple solution to your 1) potential asbestos issue, 2) leaky ductwork joints and 3) uninsulated ductwork would be to have an insulation contractor come in and spray foam the attic ductwork. That would encapsulate any potential asbestos, seal any gaps, and insulate all at once.

Stax, I agree that the white tape may not contain asbestos, but there were similar looking whitish tape products in the past made from combinations of plaster of paris, asbestos, etc. that was used as a duct wrap to seal joints.

Test it and read EPA on Duct Cleaning.

Don't Waste Your Money