It sounds like powdery mildew. Don't throw the plant out. Mildew spores are always abundant but need the right conditions to grow. Some begonias are more resistant than others so may never show any evidence of mildew. Thin leaf canes and most rexes are more susceptible to mildew. Providing lots of light, warmer air temps, and air movement are some of the proactive things to help control mildew. Fungicidal sprays are another way to help control the outbreak of mildew. A product recently touted on another website is Milstop that is supposed to be the best thing since sliced bread.

Removing damaged leafs will also help minimize the spread of mildew.

Thank you! The mildew is only on stems, not leaves, but the plant has lost a lot of leaves. I will try wiping off the stems & some fungicide. Maybe I'm also overwatering a bit. Good to know it's not transmissible to the nearby plants (like scale or mealybugs).