Our Kenyon double spreader rig is almost the same as yours and in stormy weather the inside of the spar can deliver a couple of quarts of clear rain water (in a week) into our bilge, courtesy of the interior of the spar. What with so many openings (masthead fittings, spreader roots, halyard exit plates...) there are a plethora of places for water to enter the spar interior.
I keep our bilge mostly dry by weekly visits to clean out the bilges. While a deck stepped spar can, in theory, keep a bilge dryer, I do like the stronger and lighter rig system inherent to a thru-stepped mast.
Make sure your bilge pumping scheme can get most water out; then it's up to you to dry it out with a vac or a turkey baster....
This takes constant attention, like most 'relationships' in our lives.