The existence of a password for the house is mentioned only once, and the password itself is never stated. I think there are many better examples of symbolism in this story, and that the password simply fits into the overall tone and characterization of the house rather than embodying a specific and powerful symbol on its own.
The house is mentioned as asking for the password in a somewhat oblivious challenge to the random intrusions of birds, stray dogs, foxes, etc., basically the animals that have survived the nuclear apocalypse and just happen to encounter the house by chance. This helps to characterize the house as intensely protective and conservative, but not especially intelligent; it comes across as being somewhat quaint and ridiculous in its overly-zealous mannerisms while completely lacking the ability to recognize that no one is home, or that a bird is incapable of providing a password.
If you need to specifically link the password to symbolism, it could symbolize the dependence that the house has upon human instruction and interaction, and the fact that it isn't truly intelligent.