The correct answer is B. a participial phrase modifying he.
The participial phrase is running up the stars - given that this is a participle, option A is incorrect. The subject of the sentence is the pronoun he, 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 he is doing, therefore, it is modifying it.