For the 8 years that I was diagnosed as asthmatic, I concentrated on controlling my breathing, especially during strenuous activities (i.e. hiking). "In through the nose, out through the mouth" became my mantra. I also made sure to breathe from my diaphragm and not from my chest. This allowed for full expansion of my lungs and a greater sense of controlled breathing. When I spoke with my doctor and personal trainer, they both agreed that having a strong/toned upper body helps. Good posture also helps in giving the lungs plenty of room to expand. When I went on hikes in the past, if I slouched due to fatigue, it was harder for me to breathe and catch my breath.
Here are a couple of links I found useful.
Once I got my breathing under control, I found that I rarely needed to use my rescue inhaler, even at high altitude (Colorado 14ers). As for actual gym type of exercises, I imagine any type of activity that allows for an increase in stamina will be of benefit. Weight or strength training that improves upper body muscles will most likely help too. I hope this helps.