Allergies and asthma are usually hereditary an can affect children especially if children's parents or both have an allergic disease.
These are the steps to prevent them from developing:

1. To avoid food allergies in babies, a mother should breastfeed her baby for the first to six months if this is not possible, give your babies hypoallergenic formulas like extensively hydrolyzed casein or a partially hydrolyzed whey formula instead of cow's milk or soy formula.
After this period, give the baby single-ingredient food like fruits, vegetables and cereal grains. This will help the parent eliminate any food that can cause allergic reaction. Common food that can trigger an allergic reaction are in cow's milk, soy, egg, fish, shellfish, peanuts and tree nuts.

2. Prevent environmental allergies and Asthma by reducing contact with dust mites early in life. This can help delay or prevent allergy or asthma symptoms. To avoid dust mites by using zippered "allergen-impermeable" covers on pillows and mattresses and wash bedding in hot water every week. Be sure that there is an indoor humidity below 50% and remove carpets and upholstered furniture from your baby's bedroom.

3. Be sure that your children are not exposed to tobacco smoke before or after birth. Children who will be exposed to secondhand smoke are shown to have an increase chances of developing asthma and other chronic respiratory illnesses.

Allergies can present at any time and go away at any time. Even an adult can suddenly become allergic to something, for instance I became allergic to witch hazel after I had my first son. My husband was allergic to strawberries at age 2, now he can eat them fine. I don't know much about the science of how that works, but I don't think science knows much about allergies really other than they can be hereditary.

As far as avoiding them all you can really do is pinpoint what your allergies are and avoid that, the picture you have is from an allergy test which are expensive and painful but pay if you fear a serious allergy.. like peanuts. There are some over the counter meds for allergies but for little kids under 4 most of them are not okay. I'd talk to your ped. if you have a child you suspect has allergies.