It sounds like Fin fungus but a picture would be good just to confirm.
It can look like white or gray coloured, cotton like growths on the fin and body or stringy gray tufts or fuzzy material on the fins which attacks them. The betta will normally become lethargic and may also lose its appetite. The typical treatments contain methylene blue but are normally clearly labled for fungus in you LFS. You can also use small amount of aquarium salt to help with treatment. The cause of fungus on a bettas fins is usually due to poor water conditions, parasites or injuries on the fish.