"Wet" cleaning is the most effective way of cleaning the vast majority of carpets. The reason for this in the soils in the carpet are rinsed and extracted away. The cleaning solutions used need be no more "allergic" than those in dry cleaning compounds. From experience "dry" cleaning systems are at best effective only as problem solvers, they tend to be ineffective in most situations and its difficult to get the compound out of the carpets, even with specialist equipment.
Ask yourself would you be happy washing your hair with dry granules impregnated with detergents and solvents, or would you rather use a shower which rinses the shampoo and dirt away? Why would it be any different for a carpet?