With the Broken Steel DLC, what happens if you refuse to help with the purifier? If you don't go in and you don't give the code, will that prevent Broken Steel quests from happening? I had a brain freeze and forgot to find out on this playthrough. The only save I could go to is 20+ hours ago. Who knows, I may do it anyway as I gained Karma by telling Lyons about Vault 87 (so stupid to get Karma for that) and finished the MQ with a Neutral Karma (damned Abominable Mutants) instead of evil.

You can walk away at any point, and never finish the quest. But that means the BS quests will not load up. Everyone will be waiting breathlessly for you until you decide to do it again. So really, there is no rush.

I meant when you are in the purifier at the very end of the MQ and you have the option of going into the room and turning on the purifier, giving the code to Sarah, or walking away and not give her the code. If you walk away and don't give Sarah the code will it not allow the Broken Steel quests to happen?

Yes. If you want to continue further in the BS quest line, you must go into the purifier. It doesn't matter what choice you make, but you must still make one, in order for the BS quests to load.