This is a very important question and the focus of ongoing research.
The latest update from the WHO on mobile phones and health states "A large number of studies have been performed over the last two decades to assess whether mobile phones pose a potential health risk. To date, no adverse health effects have been established as being caused by mobile phone use."
However, the WHO notes that “while an increased risk of brain tumors is not established, the increasing use of mobile phones and the lack of data for mobile phone use over time periods longer than 15 years warrant further research of mobile phone use and brain cancer risk. In particular, with the recent popularity of mobile phone use among younger people, and therefore a potentially longer lifetime of exposure, WHO has promoted further research on this group. Several studies investigating potential health effects in children and adolescents are underway”.
ARPANSA provides the following advice on children using mobile phones:
“Some parents provide mobile phones to their children for a range of reasons, including their child’s personal security or the assurance of their child being constantly contactable.
Concern has been expressed with regard to mobile phone use by children. At present, the available body of scientific evidence does not indicate that children may be more vulnerable to RF EME emissions from mobile phones than adults. The ARPANSA Standard used to regulate the emissions from mobile phones, is designed to protect people of all ages and health status against the known harmful effects of exposure to RF EME.
Although it is not considered necessary, parents may decide that they wish to lower their children's RF EME exposure when using mobile phones. In this case, parents can encourage their children to reduce call time, only make calls where reception is good, use hands-free devices or speaker options, or send text messages instead of making calls.”
http://www.arpansa.gov.au/mobilephones/index.cfm