Both of these works are set in a futuristic dystopian society in England. In both cases, human reproduction has been eliminated: In Brave New World, children are created in test tubes while mass infertility has stymied reproduction in The Children of Men. Both societies deal with the result of some cataclysmic event which has resulted in an authoritarian style of rule. Democracy is no more.
However, in Brave New World, the population does not seem to realize the mind and behavior control under which they must submit. Lifetimes of conditioning have taught them to love their world and their place in it. Nothing disrupts this stability even though some individuals, Bernard and Helmholtz as examples, internally feel ostracized for their individual feelings and beliefs.
In The Children of Men, society is much darker. People are exploited, forced into labor and routine medical exams, and the ominous presence of the State Secret Police.
For both works, the catalyst for change is the introduction, or birth, of an outsider. In Brave New World, this character is John, a child born in a Savage Reservation and brough to the new world as an adult. His disgust with the society cannot be reconciled, and he eventually commits suicide while Bernard and Helmoltz are exiled. Here, society continues as is.
Ironically, hope exists in the darker society of The Children of Men. The "outsider" here is a newly born baby. While the Omegas were once believed to be the last generation of humanity, this birth provides hope. It allows Theo to gain control with the promise of a brigher future.