A deleted scene implied he knew by second year.
Though it never appeared in the books, so may not actually be accurate to the final versions, at one point J.K. Rowling was considering putting a scene where Draco and Theodore Nott, both the sons of Death Eaters, discuss Dumbledore, Harry, and theories the Death Eaters had on how Harry survived. That implies they likely know their fathers were Death Eaters since they knew what the Death Eaters’ stories on Harry’s survival were. J.K. Rowling considered putting this scene in Chamber of Secrets, implying that (at least at one point), Draco would’ve known his father was a Death Eater by second year.
Malfoy & Nott (Chamber of Secrets/Goblet of Fire)
I liked this scene so much I tried to use it twice; unfortunately, it didn't work in either place so I finally laid it to rest in one of the cardboard boxes where I keep all my old drafts, notes, electricity bills and chewing gum wrappers.
As in the case of Dean Thomas, I know much more about Theodore Nott than has ever appeared in the books. Raised by a very elderly widower and Death Eater father, Theodore is a clever loner who does not feel the need to join gangs, including Malfoy's.
However, in this scene Theodore's father (the same Nott who was badly injured in the closing chapters of 'Order of the Phoenix') goes to visit Lucius Malfoy to discuss Voldemort-related business and we see Draco and Theodore alone in the garden having a talk of their own. I really liked the scene, firstly because it showed the Malfoys' home, and the difference between the place where Draco has grown up and number four, Privet Drive; then because we rarely see Draco talking to anybody he considers a real equal, and he is forced to see Theodore as such, because Theodore is just as pure-blooded as he is, and somewhat cleverer. Together these two Death Eaters' sons discuss Dumbledore's regime at Hogwarts and Harry Potter, with all sorts of stories that the Death Eaters tell about how this baby boy survived the Dark Lord's attack.
- Malfoy & Nott (Chamber of Secrets/Goblet of Fire), Extra Stuff (J.K. Rowling’s website)
It’s unclear if Draco was still intended to know about his father’s connection with the Death Eaters by second year after this scene wasn’t put in, but at one point, he was to be shown as knowing in second year. It’s possible that may have remained the intended truth even after the scene wasn’t added in the final book.