How about instead of calling it a negative vote we called it a counter vote. There is all kinds of strategy you can use in your voting if you so desire to help make more off your favorite movies make it and fewer of the movies you don't like.
First off, you might not include a film you love if it is an obscure one that isn't likely to get much love from anyone else (for me this might be Bend It Like Beckham...not sure that many people who have seen it dislike it, but I've not known many other people to consider it great). This would be like voting for Nader...you aren't going to win but your second favorite might lose, so you focus on those with more chance of winning.
Also, you aren't likely to use a counter-vote on just any film you dislike, because many of those probably aren't in the running for a spot either, especially at the cost of a voting slot. Thus you will tend to focus your counter-vote on something that has a good chance of making it in hopes of knocking it from first to second tier or out altogether. I just don't see what is so controversial about saying that if there is a film (let's take Crash since I like that film) where five people put it in but another five are so passionate that they'd be willing to use a counter-vote. Shouldn't this film be considered worse than a film (say Howl's Moving Castle) that seems like it could get five votes but isn't likely to draw counter-votes? This lets less divisive films prosper but also lets less well seen films prosper. If a film only has five people who have seen it in that last scenario but all of them love it, that would seem more important than a film with 20 viewers where 5 love it, 10 are in the middle, and 5 hate it.
At the end of the day, it is not about being negative but about getting a more accurate view of what is actually preferred, it just takes knowing what people don't like to do that.