Let us start by simplifying the sentence as much as possible:
You took precautions against my hearing.
The word "hearing" is a participle. By adding the suffix -ing HAL has turned the verb "hear" into a noun. This noun is the name of the action indicated by the verb "here".
HAL needs to turn "hear" into a noun because he wants to use it as the object of the preposition "against", like this:
You took precautions against hearing.
But this is not the correct meaning. Dave did not try to prevent himself from hearing. He did not want HAL to hear. So:
You took precautions against my hearing.
Since "hearing" functions as a noun, HAL can attach an adjective to it to indicate that he is the (potential) hearer. But this is not precise enough for HAL. Dave did not try to prevent HAL from hearing any sound at all, just what Dave was saying. So:
You took precautions against my hearing you.
This is good and something a native speaker would say, but it does start to strain the logic of grammar. The participle "hearing" is now both modified by an adjective "my" and followed by a direct object "you" as if it were still a verb.
You propose to resolve this tension by having HAL say "I" instead of "my". In theory this is better because it is more consistent. You construct a valid verb phrase "I hear you" and turn the whole thing into the participle "I hearing you".
And in theory this is grammatical. Native speakers would have no problem with this sentence:
Dave took precautions against HAL hearing him.
But they will tell you that your proposal is ungrammatical:
You took precautions against I hearing you.
Why? Because you have run afoul of the last vestiges of the English case system. Some pronouns still have cases! You seem to be aware of this because you have correctly chosen the nominative case form "I" for the subject of your participle phrase. But the result is incredibly jarring to a native speaker. He will likely offer the following mistaken correction:
You took precautions against me hearing you.
English speakers get little practice in using cases in difficult situations. All they have to work with is a hazy impression that after a preposition one should say "me", not "I".
This correction, which I have described as "mistaken", is actually widely accepted in conversational English. But in science fiction computers are almost always depicted as speaking formal literary English. HAL, being a computer, cannot make grammatical mistakes and he cannot use your suggestion since native speakers will see it as ungrammatical. So he says it the way he did. And a human speaking literary English would likely say it the same way.
P.S. HAL did have one more option:
You took precautions to prevent me from hearing you.
Now I/me is no longer part of a participle phrase. It is the simple object of the verb "prevent" which will now govern its case. Whether it is accusative or dative or whatever does not matter. All of the oblique cases have the form "me" and that is what we put.
