The game has a way of protecting certain "critical" NPCs that must survive in order to continue the game - they are marked as "Essential."
Most essential NPCs that follow you or lead you will defend themselves if attacked. However, if they sustain damage enough to be killed, they will simply be knocked unconscious, and they will get up as if nothing happened later.
There are some side quests that it is possible to fail if a key NPC dies, but in almost every case where the death of an NPC would prevent you from finishing the game, the required NPCs are marked as essential.
The Fallout Wikia has pages for each character, and there is typically an information box that specifies whether or not the character is essential, and under what circumstances they get or lose this attribute.
In the specific case you mention, your father is marked as essential during the trip to Rivet City, so there is no danger of him dying during this trip. (Do note that there are spoilers on that page if you haven't finished the game...)