Hugely difficult topic.
First off, the problem of victims being disbelieved or denigrated is obviously very real, and affects men just as much as women. If Jerry Sandusky's victims had had any confidence of their accusations being weighed fairly, or treated seriously, he would have been caught and tried far sooner than he was. So-called "men's rights advocates" seem to conveniently forget this when they treat the whole issue as somehow being feminism-gone-mad, and in many cases their concern about false accusation seems overblown in ways that it wouldn't be if they were taken male victims into account, which they should. Because of this, the air of misogyny that hovers over them as a result is in part their own doing.
So, how real is the problem of false accusation? False accusation is hard to track through statistics available to us because, AFAIK, nobody keeps statistical track of demonstrably false accusations; even if they did, it would be difficult to tell from such stats how much false accusation really happens. It's not even easy to tell from whether the victim's story tracks and is self-consistent or not, since being actually raped carries enough mental trauma with it to confuse recollections in ways that can make a story look false when the actual incident did happen.
Yet with enough motivation, there's plenty of reason to believe it happens, and not infrequently*. To take an historical example, it was a commonplace of interracial relationships in the Jim Crow era -- especially those where the man was black -- that the white female would claim rape if the relationship was discovered; it was easier to do so, and easier for her cohort to believe, than the alternative. (EDIT: In fact she would be heavily pressured to claim it, and it would be just as inescapably clear to her how complete and lifelong her social ruination would be if she didn't.) In the present day, the ideology of always believing the victim has had a salutary effect on making people feel safe in reporting a crime... but has also made it a fairly reliable way of ruining, or provoking violence against, someone one dislikes. And I do strongly suspect I've seen incidents of this kind of false accusation play out in real time. A couple of them involved girls who showed no sign of having been raped -- and there are signs to look for, as one can learn from work at women's shelters and in other areas of social work; a person's behaviours and sense of bodily trust with other people tend to be affected in ways that are hard to fake -- and reported the claim to circles of their friends but nobody else (and fights and incidents did indeed materialize as a result). One other involved an attempted claim against an educator who was fortunate enough that his accusers botched their attempt to construct a convincing story so badly that they had to come clean about having lied. Had they been more competent, there is little doubt he would have been ruined and forced to resign. IMO given the temptation of that kind of power, anyone who says that the false accusation theory should be dismissed out of hand is on shaky ground; experience and history teaches us that humans of any sex will be tempted to abuse any kind of power at their disposal, and this kind of power is no different.
[* EDIT: "Not infrequently" is a bit ambiguous. To clarify, I'm not someone who will cite the infamous Kanin "study" and tell you that 40 or 50% of rape allegations must be false. I'm someone who thinks that a 5 or 6% rate of false allegation, which is probably likelier, is too frequent.]
Because of the lack of tracking and how much comes down to intuition, it's hard to quantify how much of a risk false accusation really is. But it's probably worth coming down hard on proven instances of it precisely -- however few they may or may not be -- because they have real potential to poison the well for real claimants, which is a massive betrayal. Outside of this, actually observing the "presumed innocent" clause of the justice system -- while insisting that victims' claims be weighed fairly and without bias or automatic denigration -- is probably the only really constructive course of action. Anyone who insists on discarding one or the other element of this binary, no matter their motives, should be made aware that they're making a dangerous error.
As for the larger problem: educating people about what rape is and that it is unacceptable is obviously the most important step. As far as this goes, the "rape is about power" heuristic is far from the whole story but is certainly useful.
This is not to say that power and sex are readily separable; plenty of rape is about power as a part of sex, or sex as a part of power. But there's a great deal of it that is very purely about power. When someone takes the time during a home invasion robbery to rape an eighty-four year-old woman, they aren't likely doing it because it's the sexiest thing imaginable, but as an intimidation tactic. There are, for that matter, entire prison gangs built around the concept of rape-as-power (most infamously the dreaded "Numbers" gang in South Africa, but they're far from alone and their members are not selective as to sex, it demonstrably becomes just a way of communicating dominance to those who become naturalized to it [google the story of Mogamat Benjamin if you have a strong stomach and want to know what I mean]). Military and paramilitary groups in the battlefield also use it, and other forms of sexual humiliation, often in precisely this way and sometimes according to the dictates of specific, carefully-worked-out torture manuals: that's what the naked human pyramids built by the American occupants of Abu Ghraib were about. They were enacting a policy.