Chamber of Secrets

Harry enters Borgin and Burkes accidentally, on his first attempt to use Floo powder. Seeing Draco Malfoy approaching, he hides, and eavesdrops while Draco and his father Lucius are waited on by Mr. Borgin. In a particularly oily manner, Borgin suggests that it is a pity that blood purity counts for so little these days, expresses surprise that Mr. Malfoy is selling rather than buying today, commends Draco on his interest in the Hand of Glory, and finally bows the two of them obsequiously out of his shop. He then returns into the back rooms of the store, giving Harry the chance to make his way out unimpeded.

Half-Blood Prince

Harry, Ron, and Hermione follow Draco Malfoy, who has slipped away from his mother, to Borgin and Burkes. There, we overhear Draco demanding assistance in fixing something, a demand to which Borgin expresses some reluctance. Draco shows him something to inspire his cooperation. Draco also says that Fenrir will be stopping by periodically to ensure that he is doing his job. After he leaves the store, Hermione goes in to try and find out what Draco had been interested in; Borgin is uncommunicative, though he does mention that one necklace she looks at is priced at one and a half thousand Galleons. Finally he ushers her out of the shop, and puts up the Closed sign.

While we hear that Tom Riddle had been employed some years earlier at Borgin and Burkes, we do not see Mr. Borgin in relation to that employment.