What are the chances of a person with this disease passing the disease to their offspring?

Unfortunately the chances are very likely. Huntington's disease is a dominant gene so even if a one parent is heterozygous and the other one is homozygous recessive, there is still a 50% chance for the offspring to inherit this disease. If the parent is homozygous dominant instead of heterozygous there is 100% chance for the offspring to inherit the disease.

In this example one parent is homozygous dominant(HH) and the other parent is not affected at all(hh). All the offspring would all be heterozygous(Hh) for this disease. Since Huntington's disease is dominant, all the offspring would have this disease.

In this example, one parent is heterozygous for Huntington's(Hh) and the other parent is not affected at all by Huntington's(hh). There is a 50% chance of the offspring inheriting Huntington's disease in which case they would be heterozygous.