How do democracies deal with political parties that seek to abrogate democracy?

Answer 1

"Democracies" can't -- by definition they are the will of the people. Yes, some violate people's rights by banning a party or set of speaking points -- but that is a Democracy realizing that they don't really have the tools to legitimately sustain themselves.

A huge improvement, but still not infallible, is a Constitutional Republic / Democracy, where principles can put into writing, and a change requires some kind of change in principle to be encoded into the constitution. Obviously you have to enforce the principles with judges, law enforcement, an informed electorate, etc. Eventually the wrong principles can be encoded, and it is much harder to go back for the same reason it is much harder to change.

So in the end, the best defense a free country with some form of voting and free speech has is not the power to ban anti-democracy forces, but rather an educated populace which can keep educating itself and others on the logical fallacy of a people's party that is anti-people. Hopefully people can learn by logic and concepts, and not by direct experience -- obviously the latter will have learned too late that they were wrong.

Answer 2

Typically, democracies do not deal with those parties at all and simply hope that these parties will never become popular enough to cause public policy problems. This has resulted in a number of democracy-abrogating parties coming to power, such as Erdogan's AKP in Turkey, Orban's Fidesz in Hungary, and, the most ominous example, Hitler's Nazi Party in Weimar Germany.

There is a movement in some democracies to ban political parties which intend to violate the freedoms enshrined by the basic democratic process. Israel, for example, banned the Kach Kahane Chai Party which sought to make Israel a Jewish-only state in violation of the rights of Israel's Arab citizens. Neo-Nazis have been banned in several European countries and Hizb ut-Tahrir (which intends to create an Islamist Caliphate) has been banned in some countries. However, these bans or limits are few and far between.