In those sentences, at best functions as an adverbial phrase. In your examples, it modifies either an adjective or a whole clause. I've marked the part of the sentence that at best modifies in bold below:
Their response to the proposal was, at best, cool.
The government's response seems to have been at best confused and at worst dishonest.
If he drops the course now, at best he’ll get an incomplete, and he could fail.
Of course, this is English, a language where grammatical distinctions are sometimes fuzzy, and one could argue that at best modifies was in the first sentence. The main thing to know is that at best works as a whole to modify something else, and it follows the ordinary rules for prepositional phrases. Here are some other prepositional phrases that work grammatically the same way: at the earliest, at the latest, at most, at least, at all, at last, at any event.