I would consider this The Great Winter Challenge. Can you give them less moisture and more air? Some people put a small fan on them, I keep my house heating/cooling on recirculate anyway so the air moves. I spray weekly with a fungicide and try to water without touching leaves. About once a month I scrub the pots with disinfectant if they seem moldy and rinse out the pebble trays, just so nothing gets comfortable in there. It helps if they're not jammed too close together too.

Keep them warm if you can, give plenty of light, spray on a regular basis with your favorite fungicide (I have used NEEM but I don't like the smell indoors and now use Milstop), and when you see an outbreak, remove the worst leaves or all affected leaves, and you will be fine.

In the future you may want to start a spraying program in late summer/early fall to kill mildew spores before bringing indoors. If you keep them indoors and they are disease free, then as you add to your collection spray the new plants before introducing them to your current plants. I've often bought plants from a nursery and within a week they start showing ugly spots from mildew attack.

Some begonias are very susceptible to mildew (mallets are probably the worst, rexes, spotted canes, some Asian species) but all can carry the spores on their leaves and stems. If conditions are ripe (humid, cool) then it won't take long for a full attack.

My mentor says she doesn't have mildew problems but then every one of her begonias is in a terrarium. That is good for keeping pristine plants but that also means if the plant gets big then it is removed from her collection. She usually starts a new plant with a leaf or stem before getting rid of the big plant.

I'm experimenting with B. 'Weltonensis'. I've put it in a high traffic area to see if the motion of people walking by it will circulate the air and help deter mildew, it has always been mildew prone for me.