Michael,
The temperature of the exhaust gas was in the 700 degree F range, so short of someone pointing a fire hose down the funnel from the top, there was little to worry about even in heavy rain; most of the water simply vaporized long before it got much past the opening. The dummy funnel was partly covered at the top and had a network of baffles and waterways within that directed any rainwater which entered to a drain pipe which terminated at a scupper at the base of the funnel on its after side. This scupper is visible in some deck photos when taken from the proper height.
Rainwater was more of a concern when the ship was in port and the boilers beneath a given funnel were allowed to go cold. In these cases, a canvas rain tent was rigged across the mouth of the funnel. The surfaces of the inner funnels and uptakes were particularly liable to corrosion when cooled to the point where water could coat them or collect on them because they lacked any sort of protection against rust; there wasn't a coating in existence at that time that wouldn't burn off the surfaces of the inner funnel (flue) or the uptakes. Very often, the canvas cover was lightly fitted in such a manner so as to allow a gap beneath the overhanging edges to permit a bit of air circulation and a very small fire was kept burning in one or two furnaces -- just enough heat to chase away the dampness from within the network of uptakes, but not hot enough to set fire to the canvas cover, as the tops of the funnels averaged 150 feet above the tops of the fire bars.
Aerial pictures from the 1920's and 30's of the Olympic sporting such covers while docked at Southampton may be seen in a number of books.
Regards,
Scott Andrews