Old Dorset is Johnny's father, and a prominent rich man in the town of Summit. We aren't given much information about him, though Sam does describe him as:
Respectable and tight, a mortgage fancier and a stern, upright collection-plate passer and forecloser. (4)
In other words, he's kind of a jerk. Perhaps Johnny is the best indicator for the character flaws of the father. For one, we understand from the pure joy that Johnny is experiencing in his situation that his father is a tad neglectful. After all, he's perfectly willing to bluff with his son's life, as with his "counter proposition" that demands that they pay him (88). What kind of dad does that?
A pretty cruddy dad, that's what.
The other big description we get of Dorset lies in his counter offer:
It was written with a pen in a crabbed hand. (83)
By writing the letter in pen, we can gather that Old Dorset is a very confident and assured man. He has not a worry in the world about the kidnapping, and he probably has some understanding that his boy is putting these dolts through the ringer. That makes him a wise, if not necessarily good father, and of all the figures in the story, he's the only one who comes out of it with a profit.
Naturally, he's the person we know the least about, and certainly the figure we have the least sympathy for. Considering his poor parenting skills, there's a certain nod to the injustice of the universe in his final triumph. Sometimes, the jerk really does come out on top…