The correct answer is B. a participial phrase modifying he.
The participial phrase is <em>running up the stars - </em>given that this is a participle, option A is incorrect. The subject of the sentence is the pronoun <em>he, </em>not the participle. Option C is also incorrect - given that the phrase is used correctly (it modifies the pronoun), it is not dangling. A dangling modifier is used in a wrong way because it doesn't refer to anything in a sentence - since that is not the case here, C is incorrect.
That leaves us with option B - the participial phrase is describing what the pronoun <em>he </em>is doing, therefore, it is modifying it.