I have exercise-induced asthma and I have been a frequent user of DXM for almost a decade. I know that is insane to some people but I'm alive and well, healthy and in shape, although there are definitely some noticeable short term memory issues now when I do take DXM or even Gabapentin. I have never had any issues with asthma while on DXM, except when I was playing in a hockey league and used to take it hours before games, which were at night...I had some serious stamina issues and my heart was pounding like it never has before. This only happened once, and after that I reduced the dose and paced myself (and thus played poorly, so it was stupid to take, and probably damaged my arteries from the blood pressure increase).
So TLDR, no I have never had issues and I have basically had (exercise induced) asthma attacks while on DXM and the asthma attack/breathing wasn't any worse, it was more my heart.