To sell more shoes. Even in basketball where ankle sprains are very commonplace from landing on other people's feet, high-tops are no better than low-tops in preventing inversion ankle sprains.
I'm not talking about BS lab setting studies, I'm talking about the real field studies.
If you need support/protection, buy quality braces. There are TONS of different brands, styles and configurations.
El Guapo is right. High and mid top shoes don't provide any actual benefit in terms of support. There is some evidence that they may help slightly via biofeedback (essentially, the feeling of the high top acts as a tactile reminder to orient your foot correctly, but doesn't provide actual support in a rollover), but it's not much. A good medically validated ankle brace is the best course.
However, there's another factor that hasn't been mentioned on this thread. Almost everyone's feet either pronate or supinate somewhat, causing instability at the point of foot strike, and increasing the risk of rollover. One of the best preventatives is having a good solid, neutral base for your foot to strike. (Some posters did suggest particular shoes that they felt were good in this regard, but these don't help feet that aren't neutral.) This base will reduce the incidence of rollover and the only way to get it, unless you happen to have a perfectly balanced neutral foot is to get a good set of professionally made orthotics. The aftermarket versions (Superfeet, Spenco, etc.) are really just insoles and don't help at all. Go to a sports podiatry expert and get fitted with orthotics that place your ankles in a neutral position. Orthotics combined with good ankle braces are your best bet. Once you're prone to ankle problems, no shoe on the market will make an appreciable difference.
Also, if you have access to a good sports trainer, they can do a tape job (or even teach you to do one) that will outperform most braces and take up less space in your shoe.