Sam McGee sings SHUT THE DOOR on the album "Greats of Classic Country, Vol. 1," Grand Avenue CD #7004, 1999. The recordings on this album were made in the 1960s or 70s, near the end of his long career. The album credits McGee as the composer. I transcribed these few lines from a sound sample:

...sheriff up one night and says "They're robbin' me.
Twenty crooks outside my house. Please tell me what to do."
"Don't let them in," cried the sheriff. "That's my advice to you."

CHORUS: Oh, shut the door; they come in through the window.
Shut the window; they come in through the door....

This seems to make the meaning clearer: "If you shut the door, then they come in through the window, etc."