Both of these brilliant texts plunge us into a dystopian world where the protagonists are trying desperately to survive at any cost. This is the goal that drives both the characters of Offred and of the boy and his father as they variously have to combat despair, the temptation to commit suicide and the other people that they are placed with.
In The Handmaid's Tale, Offred is forced to submit to being used as a sex-slave in order to procreate for the officer she is sent to. To ensure her safety, she has to engage in a relationship with Nick due to the officer's infertility and she is also forced to kill the officer to ensure her escape. She is trapped in a world where she is forced to commit crimes and go against her better judgment in order to ensure her survival. Survival is presented as something of a luxury that you must fight to achieve rather than a right that you are given on a plate.
In the same way, in The Road, the father has to kill to protect himself and above all his son from meeting a rather grisly end. This quest for survival has transformed him from being a good man with a conscience into a naturally distrustful individual who always suspects the motives of others. Consider, for example, the father's first thought when they come across the old man by himself. At first he thinks he will be a decoy, and then, when it becomes clear that he is by himself and in a very pitiful state, note what the father says:
He looked up the road and down. If this is an ambush he goes first, he said.
The pursuit of survival at any cost has caused the father to lose his humanity, and it is his son's role to remind him of morals and values which he has long ago had to forsake.
In both of these dystopian novels, therefore, survival is shown to be something that the protagonists have to fight to achieve. Both sets of protagonists have to variously kill, lie and deceive and do things that they normally wouldn't do in order to achieve survival, which is their overarching goal.