DEET is a depolymerizer and breaks down synthetic fabrics and plastics. This may be the reason there appears to be widespread public unease about this compound. However, an assessment of the record of negative health outcomes due to application of DEET suggests the risk is quite low. In general, there have been billions of applications of DEET over 40 years of use, with very few confirmed serious problems resulting. Only about 50 cases of serious toxic effects have been identified and 75% of these stopped with no lasting effects, so scientists consider this a remarkable safety profile (Fradin and Day 2002). To put it in context, your risk of dying in while driving your car is extremely high in comparison with your risk of serious negative health outcome from applying a mosquito repellent.
Examination of the 50 serious cases lets us consider the circumstances in greater detail. Serious problems were most likely to occur when the product was ingested (swallowed) or used at high concentrations or for extended periods of time. Almost all of the negative outcomes occurred in children, suggesting that use of a product with a lower concentration (10% or less) should be considered for this age group. Some authorities suggest that infants (under 6 months) should not be exposed to DEET.
One of the newer repellents based on plant chemicals could also be considered, but these do not have a long history of use so health risks are not as clear. Products derived from plants ("natural" or "botanical" repellents) are not necessarily safer than synthetic materials like DEET. As an example, consider the dangerous natural substances produced by tobacco and the length of time it took to recognize these risks.
The Centers for Disease Control has a useful discussion of repellents: http://wwwn.cdc.gov/travel/yellowBookCh2-InsectsArthropods.aspx