Official Grammarian here. Since I am an English teacher and have taught grammar for over 10 years, I figure I better jump in.
The problem isn't with "doesn't" and "don't" the problem (as accurately reported earlier) is with "the public." Subject-Verb agreement rules state that the subject must agree with the verb. Thus the verb form of "to do" must agree with the subject "public."
You would use doesn't with a singular (for example: Ryan doesn't agree with his English teacher). You would use don't with a plural (for example: The members of the forum don't agree).
Public falls into one of those strange categories called "collective nouns." Collective nouns can be either singular or plural (examples include: audience, team, group, committee, public, . . .). For purposes of subject-verb agreement, collective nouns can be considered either plural or singular --
depending on how the writer thinks of the sentence .
Consider the following phrase:
1)The public don't know the real election issues.
vs.
2)The public doesn't vote on this until next month.
The first sentence assumes that you are discussing the public as a collection of individuals -- a situation where each individual within the group happens to fall into fall into a collective group.
The second sentence assumes that you are discussing the public as a single group, taking a group action.
I figure this was a bit of overkill, but there you have it. In short, it depends on the context in which "public" is used.
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Most Subtle Official Straightman of the Laughing Hyenas -- Grammarian Brigade.
"The only medicine for suffering, crime, and all the other woes of mankind, is wisdom." -- Thomas Huxley