The Borg weren't zombies to begin with. They were bees. Technology was their pollen. And like bees, they will ignore you unless you swat at them or threaten their hive.
However, with the repeated changes to the Borg, they became zombies, starting with First Contact. They use those arm tubules on you and you eventually become a Borg, much like a zombie would bite would eventually make you a zombie.
Though the writers could never keep things straight. Once the Borg were changed to assimilate people en masse, there's no reason why they should ignore people who "don't pose a threat." That idea is a relic from an obsolete definition of the Borg.