Right now, we have no on-screen, canonical answer to the question. And yet, it does seem to be a bit of a glaring problem. Under the circumstances, we're forced to speculate:
At the point where the decision had to be made to abandon Shenzhou, Saru would have been in charge. As we've already seen, even six months later aboard Discovery, where he is actually posted as First Officer, Saru is actually quite inexperienced with command, and command decisions. His instincts and his training-to-date would have focused heavily on the survival of his herd (that is, the crew), and not on the tactical question of leaving behind a potentially useful asset for the enemy to scavenge.
Even so, however, Shenzhou was not alone when it was abandoned. There were still several Federation starships in the area, at least one of which was presumably involved in rescuing the crew. One of those captains probably ought to have thought, "Hey, we really ought to scuttle the ship. Seems a shame, but we should."
On the other hand, it's possible that the ranking CO, whomever it was, thought that Shenzhou was potentially salvageable. They might have expected Star Fleet to immediately send in a team to bring the hulk of Shenzhou home, expecting them to rebuild and refit her and get her back in the fight. If so, Star Fleet appears to have either decided it was not worth the effort, or simply not had anyone to send. Once the war began, it appears that it heated up very quickly. Star Fleet may simply have been too stretched, and then figured it was already too late. If Kol had not decided to leave T'Kuvma's ship and followers stranded, or VoQ not been so fanatic about not using Shenzhou to get his own ship running again, Shenzhou would have been salvaged by the Klingons months before.