Alliance: please, get real. The more people there are in either faction, whining and moaning about how hopeless things are, the more likely that faction will lose. The whiners are spending all their time typing out their useless babble, not leading, fighting or healing. To boot, they're also demoralizing the whole group, breaking down all sense of coordination, cooperation, and confidence. If you encounter such a moaner/whiner, do not engage in any long discussion with them -- just put them in their place, "/ignore" them, and encourage others to do the same. They whiners are dead set on depressing themselves and anyone who will listen and no matter how witty your comebacks may be, and the more energy is wasted on them, the less is going into the game.
If it is in fact true that the Horde wins battlegrounds more often, then this is itself the reason. The "For the Horde" mentality bolsters the idea that the Horde is good at PvP. If you think your side is more naturally good at something, you're much less likely to think "Blizzard hates us!" when you lose, and much more likely to try and improve on your game for the benefit of your faction. Yes, there are plenty of Horde-side players who moan and whine too, but these don't fit in with the "For the Horde" spirit of the group as a whole -- and everyone knows it.
That said, it can be truly disheartening to lose all the time -- the whiners are having genuine feelings and those matter too. Their problem is that they wallow in these feelings and try to infect others with their mood rather than striving to improve themselves and their situation. I play Alliance as well, and I purposely guard points of interest in the Battlegrounds so that I can encourage and help direct my teammates if they don't already have a good grasp of what to do. This really isn't that hard. I'm not the best player or the best leader in the game, but what I can offer is a lot better than fruitless complaining.
I wouldn't necessarily advocate that the Alliance copy the Horde and go around saying "For the Alliance!" all the time, but we do need to try and be a little more allied than we have been up to now. The whole meaning of "Alliance" is cooperation -- we must try to show that spirit in our games if we're going to enjoy ourselves and win. We can, and should, try to familiarize ourselves with the best strategies for success in the battlegrounds, but by for the most important preparation for the battlegrounds is to conquer the voice of despair within. Defeat that, and you might just defeat the Horde.
[Edit] -- Some of the ways I've expressed things in this article have led to misunderstandings by some readers. To clarify, I'm not trying to insult anyone, or bait mindless argument here. I'm addressing a specific issue I have experienced with other members of my faction. I'm not saying that all Alliance players are whiners, nor that all Horde players are better at PvP. I'm saying that when members of my faction give up, complain all the time, and insult their teammates, they're not doing any good, and in fact are making things much much worse. I'm told that there are situations in which Horde players may complain all the time, too, so perhaps this message can apply to those individuals as well.
If you try your best to encourage your battlegroup and work as a team, then rest assured I'm not criticizing you, whoever you are: congratulations on keeping a positive attitude. However, if you're throwing about criticisms such as "All of you are the worst players I've ever seen! You noobs just suck! Alliance never wins any BGs because all of you are retards!" then you're the one I'm talking to: stop it. Everyone else: don't let these people get you down -- ignore them if you have to, but do everything you can to keep your morale up and do your best regardless of which side seems to have the advantage.