Well, first, there's the obvious problem of the child not getting enough quality sleep and the behavioral effects that brings. Second, I've mentioned above that during sleep apnea, the oxygenation of the blood decreases. It has been shown that in children with DS and heart disease, the low oxygen causes an increase in the blood pressure in the lungs as the body tries to get more oxygen. The incidence of death due to OSA is unknown. If you're unsure if your child has OSA, the way to test is through a sleep study, also called polysomnography.
Some doctors also measure carbon dioxide in the blood or exhaled air.
This is usually performed by otolaryngologists or neonatologists. Various studies have been done on children with DS, and this appears to cure OSA in a good percentage of cases, but not all. In cases where there is some concern regarding the effectiveness of the surgery, then post-surgical polysomnography is needed. Some children with severe OSA also have a narrowing of the laryngotracheal area, which explains some failures after surgery. Other surgeries to enlarge the midfacial area have also been proposed for severe cases.