Studies show that children are more likely to develop animal dander allergies later in life, if they've not had a pet or other contact with animals. Many diseases are mild in children, but much worse in adults. Some theorize that the rise in immune and autoimmune disorders are a result of living in progressively more sterile environments.
That being said, is it necessary to go out of your way to expose your children to illness? Probably not, but hand sanitizers and excessive handwashing aren't necessary either. I can't say that you are washing your hands too much, but every half hour, especially in winter, can do more harm than good. If your hands are dry or chapped, you're definitely washing too much.
I recently had a staph infection on my nose (impetigo and cellulitis) that I was having problems getting rid of (I have an autoimmune disorder), and asked the pharmacist and my doctor about precautions, such as hand sanitizer. Both recommended against hand sanitizers. Apparently, most hand sanitizers are virtually useless. They're not as effective as handwashing and they dry out skin (because of the alcohol), increasing the chance of reinfection. My ENT always tells me that skin and the immune system are the best protection against germs. Since my immune system is out of wack, I have to be more careful with my skin, but in general people don't need to really give germs much thought. Washing your hands before touching your face, he said is the best recommendation. So after using the restroom and before meals is usually sufficient.