In a lot of cases, it is easy to separate the author from the work, particularly when the author is channeling a particular kind of idea or character. If an author is pro-gay rights, but writes a book
about an anti-gay character, one shouldn't ignore the book simply because of the author's politics; on the contrary, you should read it to see how he or she deals with an alternative viewpoint. To give a more innocuous example, if the author is a pacifist and writes a novel about war, such as about soldiers, it shouldn't be difficult to separate an author's personal opinions on war from what they've written about in their books. After all, it is the author's job to write about characters and situations that may or may not be in-tune with their ordinary lives. One could even argue that an author who can channel multiple viewpoints is an accomplished one. (If someone can come up with a better example of how this works, please leave a comment.)
In that sense, I think it's silly to reject an author's work simply because of their personal viewpoint. There is value in exposing oneself to multiple worldviews, particularly since doing so means we are better prepared to deal with those we might disagree with, or might feel different from in some way or another. But, even more importantly, many authors write books that have nothing to do with their personal politics--at least, in an obvious way. Lewis Carroll, for example, has been called by many a pedophile (he wasn't, by the standards of the time, but that's neither here nor there); his work, however, shouldn't be read within that context precisely because, as far as I can remember, his work has nothing to do with his supposed pedophilia. The same is true of many other authors, dead and alive. It is also important to note that authors are not born in vacuums, or brainwashed from a young age to fit a "writer template." They come from all walks of life, from every continent and, I would hope, every country. If there is any question that the United States is the melting pot of the world, then we can all take solace in the fact that the writer's world is an unflinching melting pot.
But where I diverge from most on this particular subject is on the economic issue. Authors (and publishers) need readers to earn "money" (both in the physical currency sense and in the readers-as-currency/voice sense), and they use that money for various things. What should be of concern to anyone who cares about their own politics is that the "money" authors earn from us can also be used against our interests--here I'm specifically talking about living authors, rather than ones that have been dead for centuries. If your politics are not important to you, then you can ignore such things and continue supporting authors who would use their voices to deny things you rightfully deserve. If your politics are important, however, then it should quickly become obvious that separating an author from his or her work is an economic impossibility. Providing "monetary" gains to authors also provides incentive for them to continue doing what they do. In most cases, this isn't an issue, since many of us want our favorite writers to keep writing and talking to us. But some authors use their "money" to push their political ideologies, to speak against issues that matter to us in the most bigoted way, and so on. To support such authors is tantamount to saying "I'm okay with you using your monetary and reader-ly currency to combat my personal interests."
If you're okay with that, then you should continue buying new copies of books written by authors you vehemently disagree with. But I think that by doing so you become an accessory, which would be an insult if one could say that there is such as a thing as being not-an-accessory--which, I would argue, there is not, since we all purchase things that, somewhere down the line, work against our interests. Regardless, with authors, we have a very clear choice: we can support the ones we disagree with by giving them royalties and an audience, or we can cut them off like the diseased limb that they are and give more "money" to those who are both great writers and great people.
I've made the choice for option two a number of times, such as in the case of Orson Scott Card and John C. Wright, both of which have said many an ignorant, disgusting word about the LGBT community, and both of which are good writers. I still own books by both of them, and will continue to do so, but I won't purchase new copies of their books in the future, nor will I review their works, or provide them any sort of positive feedback which might, in some way, be used as "currency." That's something I'm not willing to do so long as their politics are so vehemently opposed to mine, and so long as they are so outwardly for the destruction of what matters to me--my family. I may make the same choice with Elizabeth Moon, depending on what happens in the next week or so. I think the more one realizes that every dollar we spend gets used for something else, including things we might disagree with, the more we have to acknowledge our responsibility, even our complicity, in the negative (subjective as it is). In the case of the separation of author and work, I think we should make the choice to step away.