What are similarities between The Handmaids Tale and Brave New World?
If you are looking for an example of how the society in Brave New World uses fear to achieve its goals, think of the end of the novel where Bernard and Watson are threatened with deportation and isolation from society. That in and of itself is a form of fear-mongering that keeps people in line. Also, think of the scene where John the savage lost control in the hospital where his mother was dying; he was immediately swarmed and subdued; that sort of militant presence is there for control through fear.
Other possible themes that are similar between The Handmaid's Tale and Brave New World is their fascination with the production of children; it is gone about in different ways in each novel, but their government has completely wrested control of reproduction from the control of the people themselves. In Brave New World it is done in factories, and in The Handmaid's Tale, they use the handmaids and an intricate system of manipulation and brainswashing to try to get babies, along with the stifling of sexual activity to discourage births outside of those guidelines. So, boths societies have complete control over reproduction.
Another similarity between these two dystopian novels is that it all started with discontentment about the way society was. In Brave New World, people were disgruntled at all of the heartache and pain that human relationships caused, and the lack of productivity in the world. From that spawned the totalitarian regime that based all of its workings on that ideology. Then, in The Handmaid's Tale, there was a think-tank put together, and they all had an ideal world where rape, violence and sexual deviancy would be done away with; they used that to form their new society with. So, there are very extreme ideologies that start a revolution that lead to the societies present in the novels.
Also, in both novels, sex is used as a tool. In Brave New World, it is a way for the masses to stay happy and satisfied, and to relieve any angst or frustration they have about their society. It is also a way to distract people, and to keep them fixated on something other than philosophical ponderings that might lead to rebellion. In The Handmaid's Tale, withholding sex and punishing people for sex is the way that they control the entire population. It has become a forbidden thing, and it is used to wield power and control.
Those are just a few other suggestions; I hope that they helped. Good luck!