When I was growing up, I was never allowed to get dirty. Germs were bad, and playing outside was considered hazardous. Whenever I had a spot on my skin that even resembled dirt, it was scrubbed until the skin turned red.
Fast-forward to today. My immune system is basically shot; whenever something "goes around", I get it without fail, and when I do get sick, it's nearly incapacitating; I have allergies to some medications, plants, and animals; my asthma (which I've had since birth) gets worse, not better; and I'm currently battling with CIU (chronic idiopathic urticaria), or hives that keep coming back without any known cause.
Please let your kids play outside. Let them get dirty. Their bodies will thank you.
Keep in mind that asthma is an autoimmune disorder that happens to manifest itself in the respiratory system. It is not in and of itself a disorder/disease of the lungs. The act of solely treating the symptoms of asthma is akin to spraying a fire extinguisher at the top of a flame, instead of at its base.
Ever since I read about the "hygiene hypothesis", I've considered infecting myself with hookworm (called Helminthic therapy) to keep my immune system busy. There are certain species of hookworm that aren't harmful to the body (under normal circumstances in first-world countries), are not hosts themselves to bacteria, viruses, or other parasites, only feed on a small amount of blood, and can neither reproduce within the host nor lay eggs while in the host. They're just enough of a nuisance to prevent the immune system from attacking random things in the body, and they work around the clock.
The only reason I haven't actively sought out this therapy is due to my primary care physician's recommendation against it. He grew up in a country where hookworm is common, and his sister died of an asthma attack; the fact that she most likely had hookworm and had asthma severe enough to kill her was convincing enough to me that this research may not be completely accurate.