I am mildly asthmatic, although my asthma has been really well-controlled since I became a serious biker. That having been said, I do, occassionally, get that raw, irritated feeling in my lungs and bronchi from cold weather cycling, especially after more strenuous rides.
I do two things: (1)I use a balaclava hood under my helmet (like a robber's mask...covers nose and mouth....only eyes exposed).
(2) I drink warm herbal tea after most cold-weather rides.
Here are the +/- I have found with the balaclava.
+ For me, it's more comfortable than a scarf. It stays in place. Also, and most importantly, because it covers the whole head, neck, nose....essentially, only your eyes are exposed, I rarely get that irritated feeling in my chest. Many are made of high-tech fabric that is both warm and wicks away moisture.
- Sometimes it's too hot, and often, my goggles fog. If you pull it off of your nose for a while, things clear up, but it's hard to go full on and not steam up the goggles.
I've gotten really good deals on balaclavas through both Performance Bike and Sierra Trading Post in the past (<$10).
Mary Ann