Perspiration will indeed damage the leather - after many happy years of riding. I would say that covering the saddle with a waterproof/sweatproof cover totally negates the greatest benefits of using a leather saddle altogether.

Those being:
1) Moisture (sweat) wicking away from the rider's skin, which leads to a very real reduction in sores.
2) ...which results in a drier, frictionless ride - another sore preventive characteristic of a solid leather saddle.

Reasonable care and treatment of your Brooks will keep it healthy for a good twenty years or more. Use fenders, cover it while parked in the rain, and use sensible leather treatments periodically. If you don't like fenders, use a generous coating of SnoSeal on the underside only. Apply it a couple times during winter.