The balance of character traits with which Richard Hughes endows Captain Jonsen effectively prevent him from being any sort of hero. His downfall is not a tragedy. Far from being heroic, Jonsen is a child abuser, a liar, a drunk, and a thief; a pirate by definition is a person who makes their living raiding other ships and stealing other people's property.
A tragic hero, in the classical Greek sense, is a noble character who has one flaw or who is the victim of fate in circumstances that they cannot control: the quintessential representative is Oedipus. Instead, Jonsen lacks a strong moral core. Although he does not kill the children, he takes no steps to deliver them to safety. Instead, he sexually assaults Margaret and leaves Emily in the cabin with the Dutch captain whom she later kills—a situation that leads to her mental collapse and almost gets Margaret killed.
As the question implies, it would be a mistake to describe Captain Jonsen, the commander of a pirate ship that inadvertently takes on the Bas-Thornton children and their two Creole companions while in the process of seizing a British merchant ship, as a tragic hero.
The captain, like much of his crew, is hardly the vicious force of nature his occupation would imply. His attitude toward the children is a strange mixture of callous indifference and desultory compassion. Everything he does seems to partake of this dual nature, his behavior a repertory of half-gestures. When, in his growing fondness for the ten-year-old Emily, he begins stroking her hair in a muted sexual overture, he's ashamed, rather than angered, when she bites his thumb and runs away.
Like the unfolding action of the novel itself, his character is governed less by its own imperatives than by chance and circumstance. Thus, when Emily falsely accuses him of the murder of the Dutch captain, it seems less an act of revenge than a randomly cruel twist of fate.
Aristotle's definition of the tragic hero states that the hero must experience an "anagnorisis," a recognition that, due to an error in judgment, they are the cause of the terrible reversal of fortune that they have experienced. Such recognition implies high character, intelligence, and, in general, a nature sympathetic enough to evoke fear and pity.
Hypothetically, in order to fulfill this role, Captain Jonsen would have had to display an awareness of the suffering he had wrought in kidnapping these children, including the death of one of them, and would have had to turn himself and his crew in to the British authorities. But he never truly experiences such an anagnorisis, and thus, he cannot be regarded as a tragic hero.
A tragic hero is a character who, despite possessing some admirable or heroic qualities, experiences a series of misfortunes that lead to a final downfall. A tragic hero usually has some flaw or idiosyncrasy that influences events, but the tragedy might also be the result of forces beyond the character's control.
In A High Wind in Jamaica, Captain Jonsen's interactions with the Bas-Thornton children, combined with his acts of piracy, bring about his downfall. His affection for Emily leads him to bring the child to his cabin when she is injured, which sets off a chain of events in which Emily is left alone with the captured captain of a Dutch vessel and kills him when he attempts to reach for a knife and escape. Jonsen's efforts to remove the children from his ship and find them safe passage also backfire, as Emily reveals the pirates' identity and testifies that Jonsen committed the murder of the Dutch captain, which leads to Jonsen being hanged for a crime he did not commit. Jonsen's tragic flaw is that he fails to recognize that the children are not entirely innocent and are capable of savage acts.