Dalmatians, like any breed, have specific problems for which reputable breeders screen.
All Dal puppies should have a BAER hearing test as a young puppy (5-12 weeks old) as some Dalmatians can be deaf in one or both ears. This test should be done before pets are placed in their future homes, and the results should be submitted to OFA.org as well as a copy given to new puppy owners.
Most Dalmatians are unable to form one ingredient in urine normally and therefore form urate urinary crystals, but a few carry the gene which prevents this problem. The technical name for this is hyperuricosuria. Dogs which have the enzyme have the "low uric acid" or "normal uric acid" (LUA or NUA) version; most Dalmatians have high urate (hu) form and are at increased risk of forming stones.
As with many breeds, Dalmatians should get their hips, and sometimes elbows, screened for dysplasia. Some bloodlines have OCD and should get shoulder radiographs. Eyes should be checked for iris dysplasia and distichia, along with any other eye disorder. Thyroid screening should be done. Some Dalmatians have Copper Storage Hepatopathy; this can be managed if identified early. Dogs from affected bloodlines should be screened regularly.
Breeding dogs need to be periodically screened for Brucellosis and, if indicated, other sexually transimitted disorders.
Your breeder can give you copies of test results. In the US, reputable breeders list their results online: click here: Orthopedic Foundation for Animals. Dogs which have completed OFA's recommended minimum health tests will have a CHIC number to make it easier to find their results.
There are other, uncommon disorders. Some genetic disorders (ARDS, Degenerative Myelopathy, etc) can be avoided with DNA testing. Some do not have a DNA test (early kidney failure, seizures, copper storage, etc) and are minimized with careful breeding.