I've always battled respiratory ailments, yet I wanted to ride all winter and early spring (our rainiest season). So my first bit of cyclist clothing was a Shimano Storm Jacket. It's been warm enough into the high 30s with just a long sleeve cotton pullover, and dry through some rainy rides.
On the down side, even with the rear cape-type vent it can get sweaty at temps above 60F. So my next purchase, around January or February, was a Pearl Izumi base layer undershirt, and a couple of poly short sleeve casual fit cycling jerseys. Much more comfortable than cotton under the Shimano jacket, tolerable even up to 80F for short rides (a couplafew miles at a leisurely pace).
Medical researchers keep telling us we don't get colds, the flu, etc., just from being caught in cold wet weather. But our bodies say otherwise. It may be that the congested sinuses from warm/cold/warm again temps are better breeding grounds for respiratory bugs.
My experience tells me I don't get respiratory bugs nearly as often when I'm warm and reasonably dry -- although sweaty is okay, as long as I don't get chilled.
Keeping the nasal passages moist may help too. I wore a paper dust mask, bandanna or scarf over my mouth and nose on a couple of dry cold rides. Seemed to help, including reducing the asthma attacks. On rainy chilly rides I don't cover my mouth and nose, but we didn't get many seriously freezing days this winter.