That is always a key question which can only be answered in the context of a particular struggle by those organizing that struggle. There is no general theoretical answer since it depends on particular circumstances.
But for example, those who already agree that public school teachers should not be hassled and should be secure in their jobs. I don't know what the percentage is who feel that way, but I know it is well over 10% -- more than enough to mount quite an effective struggle. But one has to provide them with sufficient information, for example, not to be embarrassed when chatting with others on the topic. On this particular question it would probably be sufficient to say such and such an expert says so. What counts is the intensity of their feeling about teachers; the rest can be fairly casual. If a campaign grows, the very growth will attract people who can prepare propaganda (internal education). And so forth.
Carrol