Background Down syndrome (DS) affects not only children but also their families. Much remains to be learned about factors that influence how families of children with DS function, especially families in non‐Western populations. The purpose of this cross‐sectional, correlational study was to examine how family demographics, family demands and social support relate to family functioning as well as the potential mediating effect of social support on the relationship between family demands and family functioning in Taiwanese families of children with DS. Method One hundred and fifty‐five parents (80 mothers and 75 fathers) from 83 families independently completed mailed questionnaires. Data were analysed using a principal component analysis and mixed linear modelling. Results Families having older children with DS, greater parental education, higher family income, fewer family demands and greater social support contributed to healthier family functioning. Social support partially mediated the effects of family demands on family functioning. Conclusions Family demographics, family demands and social support appear to be important factors that may play a critical role in how Taiwanese families respond to the birth of a child with DS. Care of children with DS and their families is likely to be more effective if professionals working with these families are aware of factors that contribute to healthy family functioning.