The problem with trying to win people over regarding anything - issue, product, service, cause, etc. - is this:
No amount of information changes a belief.
It does not matter if you have the most prominent climate scientists in the world, or how many; if someone believes that climate change is fabricated, they will continue to believe that. If you believe that black is white and white is black, no matter how many flash cards you are shown, you will never see what anyone who does not share your conviction sees. Belief is that powerful.
Why? Because belief feeds pleasure- and comfort-inducing centers in the brain, even for those who tend to be thinkers rather than believers.
Basically, humans who believe something that is not based on fact are those whose cognitive mind is based more on intuition than on reflection. That is, they are more apt to 'skip ahead' to answers that require less conscious thought. For reflective minds, a series of easily confirmed facts presented by a source whom the observer recognizes as an authority is often enough... it's just a matter of getting your message out there. But for 'believers', even recanted statements and authoritative proof are not enough (witness Toyota's debacle with unintended acceleration). And because most people tend to congregate with others who think as they do, this confirmation of belief is constantly reinforced.
One of the positives of social media is that it is easier to directly interact with Customers than ever before. But one of the negatives is that, if a proportion of your Customer base is intuitive rather than reflective (and they are), a comment by someone who is not a fan of your organization on a blog, Twitter, or Facebook has the potential to drive away large numbers of other Customers - even if what they say has no facts to support it.
So what can influence belief in a positive way?
A better, 'hookier' story (that is, one that is more 'intuitively true') and a peer group whose ratio tips at least 70% in favor of the new belief, so that the believer feels that they are part of an easily recognizable majority. (This does not mean the majority of humans - just the majority of the people with whom they spend the bulk of their 'voluntary' time - that is, time away from work.)
The first part is relatively easy, but how do you encourage people to regularly interact outside their usual (like-minded) peer group? Especially when social media applications - for purposes of marketing demographics - keep herding us into ever-smaller, more narrowly defined 'communities'?
The first step is to identify which of your Customers are reflective and which are intuitive. This is easily accomplished with surveys, monitoring of Twitter, Facebook, industry-oriented blogs, etc. The next step is to state the facts in language that is appropriate to each group (the days of "one message fits all" are gone), and in forums that identify your organization as part of or affiliated with the group in question (obviously, not if they are a hate group or part of some other objectionable organization). This is most easily accomplished by explaining how your organization has supported Customers like the Customer whom you wish to reach, using examples that have value to that particular Customer segment.
Bear in mind, if your Customer base has a wide demographic variation, you may have to craft several different versions of your story to communicate it most effectively to each group. It also never hurts to enlist the help of prominent members of each group to help explain your position and to learn more about the concerns of each Customer segment.
At the end of the day, it is always better to build your base of support before something negative is said, so that your Customers already communicate with you regularly, consider you 'one of them', and are more apt to listen to your side than to someone who has no facts to support them. The most important thing, if something negative is said, is to answer instantly, invite the speaker to discuss the issue off-line, listen carefully to their issue without interruption, respond in a way that actually addresses their concern, and then ask for feedback.