When the Lone Wanderer returns to the vault in Trouble on the Homefront, Wilkins angrily talks about how much he despises them and their father, blaming the two for the dire situation Vault 101 is in after they escaped, saying, "You never should have left, kid. And now we'll make sure that nobody ever leaves again!" This is most likely due to the deaths of his two children during the vault riot.
pc If you go down to the basement sub-level during Trouble on the Homefront and enter the room where the Wanderer had to shoot the targets as a child, where you encounter Officer Wilkins, you may find that opening the door and dropping a mine immediately would prevent officer Wilkins from entering his "you should have never left kid" dialogue, keeping him friendly. [verified]
pc After opening the door, it is possible to see Wilkins spontaneously teleport back a good distance from the player character. [verified]
ps3xbox360pc Wilkins tends to be located within the path to the Overseer's office, and can automatically turn the Overseer hostile if he is near. [verified]