Like Sir Charles before him, Sir Henry is more of an excuse for the novel's plot than a fully developed character in his own right. Watson comes out and tells us that Sir Henry is a true "descendant […] of that long line of high-blooded, fiery, and masterful men" (6.26). Watson is pleased that, "this was at least a comrade for whom one might venture to take a risk with the certainty that he would bravely share it" (6.26). In other words, Sir Henry's worth all of this trouble from Holmes and Watson because he's (a) a good guy, (b) a brave fellow, and (c) connected to an old English family that deserves respect. Watson thinks he's a dashing fellow, right down to the "sensitive nostrils" (6.26).
Not only is Sir Henry brave and gentlemanly, but he's also a romantic at heart. He quickly falls for Beryl Stapleton—partly because she appears to be the only eligible woman in the area—and asks to marry her. Sir Henry's warm-heartedness and his confusingly bad treatment from Stapleton (before, at least, we know that Stapleton is trying to kill him) win yet more audience sympathy from the readers. Especially those readers with a soft spot for nostrils.
In some ways, Sir Henry seems to have all the luck. After all, he inherits a whopping huge fortune, more or less out of the blue, so that he can leave behind his farm in Canada and come to England to become the lord of Baskerville Hall.
But Sir Henry's good luck doesn't change the fact that he's got a scheming murderer after him and that the one girl to whom he's given his heart happens to be married to that murderer. So we're pretty happy for him when he doesn't actually end up getting eaten by the Hound of the Baskervilles. But we're bummed that he does wind up having to go on a long trip with Doctor Mortimer to relax and recover his nerves after his terrible experiences. We hope he likes talking about skulls…