It's not about indifference or apathy to the world but about people who just do what they are told, never question why they should do that.
You've nearly answered your own question, as the word unquestioning conveys exactly what you describe:
Accepting something without dissent or doubt.
ready to accept control or instruction; submissive.
"a cheap and docile workforce"
synonyms: compliant, obedient, pliant, dutiful, willing, passive, submissive, deferential, tame, meek, mild, lamblike, unassertive, unresisting, yielding, cooperative, amenable, accommodating, biddable, persuadable, ductile, manageable, controllable, tractable, malleable, manipulable, easily manipulated, easily handled, like putty in one's hands;
That type of person can be described as a Yes-man:
A person who agrees with everything that is said.
One who endorses or supports, without criticism, every opinion or proposal of an associate or superior.
People who do what they're told without questioning why are sometimes referred to as Sheep: an easily led person.
That person is Obedient
Obedient - Dutifully complying with the commands, orders, or instructions of one in authority. http://www.thefreedictionary.com/obedient
If you want to emphasize the "do what they're told" aspect, you could say "subservient"