The original Kilz is 50 years old. Paint technology and chemistry has come a long way since then. Kilz has a dozen different types of primer now, some general purpose, some with specialized purposes. They make a "Kitchen and Bath" primer that is for high humidity areas, preventing mildew, and promoting adhesion. I have not used this particular primer, but in this application, it sounds perfect. I have used many of their other primers and think highly of them. ( except the Original.....)
Also, in this application, I'd consider an "interior/exterior" paint instead of an exclusively "interior". Exterior paint is more mildew and moisture resistant and can withstand temperature extremes and sunlight better. However, it tends to offgas for a considerably longer time, and isn't good for an enclosed space for that reason. Int/Ext is a compromise between the two. You get the greater durability without the off gassing. 1 Caveat: If you have kids who will be writing on the walls or you find yourself scrubbing walls often, go with the interior.