Bassett is the "young gardener" who works for Paul's family. He is "a shortish fellow with a little brown mustache, and sharp little brown eyes." Basset had been "wounded in the left foot in the war," and is an afficionado of horse racing. He keeps Paul's secret because of the relationship of mutual respect which he has developed with the boy, and also because of his keen awareness of his place as a servant in Paul's family.
Bassett is the only adult in the story who treats Paul with respect. He takes what Paul says seriously, follows his instructions on how to bet on the horses, and keeps the boy's winnings safely hidden away for him. Bassett is trustworthy, and he keeps the transactions between himself and Paul a secret because he believes the boy would want him to. He shares the secret with no one, until asked about it specifically by Paul's Uncle, Oscar Cresswell. In Basset's eyes, Cresswell is his employer and authority, and so someone who also must be respected and obeyed.
Bassett's awareness of his station in life, coupled with his respect for authority, is another reason why he keeps Paul's secret as far as he can, but it is also the reason why Cresswell eventually finds out about the success the two of them are having with the horses. Because Paul is a member of his employer's family, Bassett undoubtedly feels a sense of obligation to keep his secret, but by the same token, he also feels that he has to reveal something of the secret to Paul's Uncle when he inquires about it, because Paul's Uncle is his actual employer. Bassett tries, out of duty but mostly out of respect, to remain faithful to his perceived belief that Paul would want knowledge of his almost magical intuition with the horses to remain secret by telling Cresswell diplomatically,
"I don't want to give him away - he's a young sport, a fine sport, sir. Would you mind asking him yourself? He sort of takes a pleasure in it, and perhaps he'd feel I was giving him away, sir, if you don't mind."