Clearances:
With the advancements made in research, it is now possible to have your Basset tested for Thrombopathia.  This is a bleeding disorder specific to Basset Hounds.  Many breeders now test for this, as it is a definitive test. Glaucoma is probably the most known disease that affects our breed. The exam for this is not definitive, but it is used as a guideline for breeding.  The test is called a gonioscopy and evaluates the angles within the eye. If there is a high percentage of closure, research is beginning to indicate that this increases the chances of the dog developing glaucoma.  This is a subjective test at this time, but a starting point to help keep the occurrence of this disease to a minimum.  Hopefully in the future it can be elimated wtih selective breeding. 
For more information on health issues, check out the
 Basset Hound Club of America website.