Should Christians study philosophy?

Paul interacted with some Athenian philosophers in Ac 17:

18 Some of the Epicurean and Stoic philosophers also conversed with him. And some said, “What does this babbler wish to say?”

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They were rather dismissive about Paul. Others were not:

Others said, “He seems to be a preacher of foreign divinities”—because he was preaching Jesus and the resurrection.

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Paul expounded Jesus and God to his hearers. In the end:

32 Now when they heard of the resurrection of the dead, some mocked. But others said, “We will hear you again about this.” 33 So Paul went out from their midst. 34 But some men joined him and believed, among whom also were Dionysius the Areopagite and a woman named Damaris and others with them.

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The results were mixed. The philosophers probably resisted Paul's argument more.
More than a decade later, Paul wrote in Col 2:

8 See to it that no one takes you captive by philosophy and empty deceit, according to human tradition, according to the elemental spirits of the world, and not according to Christ.

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through (by)
διὰ (dia)
Preposition
Strong's 1223: A primary preposition denoting the channel of an act; through.
philosophy
φιλοσοφίας (philosophias)
Noun - Genitive Feminine Singular
Strong's 5385: From philosophos; 'philosophy', i.e., Jewish sophistry.
BDAG φιλοσοφία:

philosophy, in one pass. and in a pejorative sense, of erroneous teaching Col 2:8 (perhaps in an unfavorable sense also in the Herm. In 4 Macc 5:11 the tyrant Antiochus terms the Hebrews’ religion a φλύαρος φιλοσοφία).

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what kind of genitive usage was φιλοσοφίας?
When used with the genitive, διὰ often indicates means or instrument ("through" or "by means of"). It was a genitive of means to an end (being taken captive) rather than just the instrument. It also served as a descriptive genitive. The philosophy was paired with empty deceit (another genitive). Paul wasn't talking about philosophy in general. Philosophy was not inherently bad, but empty and deceitful philosophy was.
How do you justify ASV translating it to "his philosophy"?
διὰ τῆς φιλοσοφίας καὶ κενῆς ἀπάτης
It wasn't a genitive of possession. The article was there. I would not try to justify ASV translation grammatically.
Why is his philosophy the correct translation?
It is not.
isn't it just a genitive because of διὰ?
It is that, but not just that.
Is it clear from just the Greek that that type of Philosophy is also empty deceit?
Right, by pairing two genitive nouns with καὶ.
Should Christians study philosophy?
If you like it, yes, but don't use philosophy to deceive people. Paul wasn't condemning all philosophy but rather warning against deceptive philosophical teachings.