I'm also asthmatic, and besides the occasional flair, it's mostly exercise induced. Drives me crazy. Sometimes I wonder if the tightness and feeling like I'm going to suffocate is all in my mind. I've never, in my life, been physically active. Having to exert myself is new, and I don't like the feeling of being uncomfortable. I'm sure those who have always been active or were active in their younger years have adapted to that feeling of discomfort while exercising hard.
It does get easier if you stick with it. If you really want to run, you just have to set a goal. I hear Couch to 5K is good plan. I did my own routine to build up my time based on this couch to 5K. It took a lot longer, but when you're lungs are on fire, it helps to build up more slowly. IMHO.
I say I run, but it's really more like a jog. I'm a slow runner. I have come to enjoy it more than other exercises, and it's my preference for the moment. Who knows what I'll end up doing next, but right now, running is my thing.
If it's really not your thing, I hear this Nordic Walking is a really great workout, and burns the same calories as running in about the same amount of time. I'm not sure if that's true or not, but it's something worth investigating.