how did the democratic party let itself be labeled as...

[Liberté]

Because it allowed itself to be labeled, and in this case far more accurately, as the Party of bureaucrats. And as most of the social progress of the 20th century came by way of the government (even though a lot of this progress actually occurred under the Republican administrations of the mid-20th century), the Democrats became tarred with those labels in the process. There's no reason for a Party of "feminists, fags, pointy headed intellectuals, lazy inner city minorities" and "evil godless liberals" to be married to the government. Their natural role is in opposition to the majority, and consequentially in opposition to the government of the majority.

[Liberté]

You're new to politics, aren't you . . . .

Well, for starters, the Republicans have already been labeled as the party of Ignorant Racist Greedy Evangelicals who want to burn gays and murder abortion doctors while swimming in their vaults of gold pieces they got from denying Orphans medical care.  It's just a matter of where you live and who controls the messaging system there--whoever's in control sets the tone.  There's a reason the Republican party is fading in the Northern Suburbs they used to get 60% in after all.

That doesn't really explain how those labels got applied to each separate Party. If we analyze it historically, we find that the Democrats became associated with "Big Government" around the time of the New Deal. When the Supreme Court began desegregating schools and allowing legalized abortion - though both occurred under Republican majorities appointed by Republican Presidents - the Democrats got stuck with the nomenclature of not only being in favor of an activist government economically, but in favor of a government that would 'impose' what Democrats call equality and what I call freedom on the rest of society.