A, B - Incorrect: The modifying phrase that comes at the end of the sentence modifies the preceding nouns mukluk boots and parka hoods, which are the subjects of the sentence. The modifier a method does not make sense in describing the clothing items. Answer choice B incorrectly uses the preposition as, which implies that items of wear are a method.
D - Incorrect: Referring to parka hoods as a protection is very awkward.
E - Incorrect: Infinitives such as to protect should not be used directly after a comma.
C - Correct: The active participle protecting correctly modifies mukluk boots and parka hoods to convey what these clothing items do for the people of Prudhoe Bay.
For anyone who’s curious, the temperature in Prudhoe Bay, Alaska is -21 °C (-6 °F) at the time of this post. That’s slightly warmer than the average temperature in Prudhoe for this time of year.