As a native (American) English speaker, they both sound correct to me. For more in-depth grammatical analysis, read on. If anyone finds any errors, please point them out and I will attempt to fix them.
He would not be averse to you smoking a cigar.
This is a more construction construction, at least in my experience. It appears to be a usage of the present participle. Just as a subtle connotation difference, in this example the thing that "he is averse to" is "you smoking a cigar".
He would not be averse to your smoking a cigar.
This construction sounds higher-class, possibly more literary to me. It is definitely not as common, but I believe it is still completely correct. It is a usage of a gerund. Were I to use this phrase, I would add an "of": "He would not be averse to your smoking of a cigar", but I belive it is also correct without the "of"; I would only add it because I think it flows better. The connotation difference is that the thing "he is averse to" is "smoking a cigar", which action just so happens to be yours.
In reality, there is no real difference between the two phrases, and both are correct, although the one with "you" rather than "your" is more common.