I don't have a toddler yet but I used to take care of my niece who is now 4 years old. I would suggest that swimming is one of the best sport for kids with asthma. because it strengthens their lungs. Aside from that, toddlers loves to swim and playing in the water. Swimming will be learn and play for them. Learning to follow instructions and obeying instructors. Playing and fun as they will stay in the water for a long time. In this, they will also appreciate their parents more as they will think that their parents attends on what they love to do.
Usually kids with asthma avoid any sport of activity that requires long endurance with little resting time. Sports like golf and gymnastics are good choices. My best friend is asthmatic and she competed at a regional level in her country with no problems. In fact, she says that gymnastics has helped her with her asthma by strengthening her lungs.
I have a 4yr old with asthma and she's very fond of swimming. She's not having trouble with it. ever since we frequented our clubhouse pool, we noticed a big improvement. :)
Swimming would be best for your child. It will help develop his/her lungs well.
Angel Yaba-Dulce