There is a great likelihood that asthma is hereditary. 

There is a greater likelihood of asthma occurring if a person has a parent who suffers from the respiratory condition. At the same time, though such a person is three to six times more likely to develop asthma than an individual without a suffering parent, scientists have yet to link a single gene to the condition. ["Is Asthma Inherited", mayo clinic]

Other Triggers

Children with a particular lineage to asthma or medical history that puts them at risk of respiratory complications can see their chances of developing asthma increased by other factors. Over a seven-year-period in Cincinnati, nearly 200 young subjects were monitored. Research concluded that children who were exposed to mold in the homes or apartments where they were raised were three times more likely of developing asthma.

Mold in residences comes mainly from water damage. So children raised in the aftermath of a natural disaster such as 2005's Hurricane Katrina, especially those who belong to families to poor to properly relocate or renovate, can be placed at greater risk of developing asthma. ["Household Mold During Infancy May Trigger Asthma", US News and World Report]

Associated Costs

The treatment and lost school days caused by cases of childhood asthma can be far greater in cases where the condition is poorly dealt with during initial phases.  Over a period of several years, the difference between a child whose asthma is well-controlled and poorly controlled can be as high as seven or eight thousand dollars.

The study of 628 children ages six to 12 also found a tremendous discrepancy between these two groups when it came to lost schooldays. Kids with very poorly controlled asthma missed an average of 18 days of school per year, while other groups logged an absence of only two yearly schooldays. ["Poorly Controlled Asthma Doubles Costs", United Press International]

Air Filters

One way to mitigate the situation, for all children, is the use of an air filter in households where one or more people smoke. A controlled experiment with inner-city kids in Baltimore found that when a pair of high-particulate air filters (HEPA) were placed in the living room and child's bedroom, that the equivalent of 33 asthma-free days per year was derived. ["Air Cleaner Helps Asthmatic Kids Living with Smokers", Reuters]