The necktie has (almost) always been seen as a piece of cloth that communicates about you, an instant image (cliché?) of who you are and to which part of the society you belong to. From mercenary to nobility, it sets you up as "part of this club". From the material to the type of knot, it's intended to say a lot about you at the very first sight, so that the person meeting you for the first time knows who they're talking to. 1. Do we live in the same world? 2. Are we part of the same club? 3. Is it worth talking to you?
At the time of the movie, not everyone would wear a tie. Only upper-class people would on a daily-basis, no worker would have one, maybe except on sunday, to go to church/wedding/funeral.
"Never belong to any club that would have someone like you for a member."
Staining one's necktie would probably means: "you don't deserve being part of that club (anymore)". It's picking on you and making fun of you at the same time, while embarassing you a bit maybe. Intended to be funny, a joke played on you, not very surprizing when quoting Groucho Marx...
The screen writers guild was also a union when it was created, so it can be seen as a labor vs employers club. Staining the tie could be interpreted as "you are just a slave to management as you wear the same outfit, and that's what you deserve, what we will do to you".