They have never said what breed Black Beauty was, but with my knowledge on horses I'm thinking he might be a Frisian.
The movie started with them in Arabia. Considering that is where black beauty came from, you would have to assume that he is an Arabian horse. As for whoever posted the answer before mine, Frisian's are much bigger through the flanks and their tails and manes are much bushier.
EDIT:
^Incorrect, Friesians have "feathers" long hair on their legs and they did not originate from Arabia. They originate in Friesland a country in the Netherlands.
Black Beauty was played by a Quarter Horse named Docs Keepin Time.
Jerry Baker is a kind and compassionate character in "Black Beauty." He is a skilled groom and takes good care of the horses he works with. Jerry also treats Black Beauty with respect and understands the importance of treating animals well.