I always find that the best thing to do is, keep everyone on every second square, behind the spectral keeper, sort of in a triangle shape (I wish I could draw a diagram - I'm hopeless :P). This way his back is always to you and his evil sweeping attack will bypass you. When the little platforms start glowing, quickly switch your character to another, safe platform, and then switch them back. When the platforms glow should be the only time that you break the triangle.
The Spectral Keeper will change direction too, so move around the squares carefully to maintain the triangle. Have Yuna in your party to cast Esuna, because when the Spec. Keeper hits a character with Berserk (whilst it makes the character stronger), it also makes the Keeper angry, and you don't want that.
The first time I played FFX, I found the Spectral Keeper really hard too, but the second time I played it, I beat the Spec. Keeper in one go. It's all about strategy. Also, I find that Berserkproof armor, overdrives, aeons, Haste and Remedies all work an absolute treat in this battle. I really hope I helped.