Genre awards come in two types: public/member voted awards (which are really just popularity contests) and juried awards (which for me carry more weight as barometers of quality). And as long as you appreciate "best" does not necessarily equate to "most popular" there's no real problem.
As voted awards measure what is "most popular among the electorate", there's no reason to expect the voters to read the entire year's published output and make qualitative decisions. They're just voting for what (or who) they like. These kind of awards work best when there are enough votes so that any skewing factors (such as attempts at log-rolling, or simply authors with sizeable groups of friends) cease to be statistically significant. They run into problems with small electorates. On the other hand there is less work involved in administrating them. Juried awards need much more organizing in terms of selecting the jury and making sure that they receive all the eligible materials and attend the required number of meetings. And the credentials of a jury are always liable to be questioned in some quarters.
Personally, I like the BFS as a member voted award, but the numbers of voting members need to be increased. That's not just a case of increasing the membership of the society, it's also about encouraging enfranchisement to vote across the existing membership and making it as easy as possible for them to do so in good time. For instance I really admire the advances the BSFA have made in recent years with publishing the shortlisted short stories (and engaging with Starship Sofa to produce audio versions) and reprinting the eligible artworks too. This way, members *can*, if they so choose, make a judgement on the basis of looking at all of the contenders (yes, this is more difficult for other categories, but that's no reason not to do it for short story and artwork).
The other thing I was thinking - I don't know how many people vote for the BFS awards, but my gut feeling is that the numbers aren't huge. Nevertheless, in the instances of accountability I think that the voting figures should be published - if not publicly, then at least for the membership to see. That's the only way to avoid stooshies like the one that's currently ongoing.