How was gene therapy able to cure diseases through the transformation of actively dividing cells?
Short answer
Both dividing and non-dividing cells can permanently be modified with gene therapy by using vectors that incorporate their payload in the host's chromosomal DNA  

Background
Vector-based gene therapy can be applied by using vectors, such as modified retroviruses or adenoviruses, that incorporate their payload in the host's chromosome. 

Whether the transformed cells are replicating yes or no, the parent as well as any possible daughter cells will carry the new DNA. For example, immunodeficiency has been treated successfully with gene therapy, but the first trials resulted in blood-bone cancers, because the stem cells started to proliferate uncontrollably, hence dividing cells definitely can be permanently transformed. In fact, when germ line cells are transformed in this way, even the offspring of the host will inherit it.    

Sources
- American Medical Association
- Learn Genetics
- Utah University