The Royal Family has guards, definitely, you see them in a lot of the games. Of course, they're also shown as comically incompetent in some of the games, like OoT's Castle Guards just wandering around in circles in the courtyard and failing to keep a kid out, or TP's guards whimpering at the sight of a single wolf.
Comical ineptitude aside, they were also going up against Ganondorf or other similar evil forces. Zelda's guards aren't strong enough, and it's not her fault she's usually the target of the most powerful villain in the land. Of the ones we see, most of them get killed off (you even see one dying in OoT's alleyway). I mean, in TP she practically had a whole army defending her before surrendering, and they still got torn down by Zant and his magical Twili army. Normal bodyguards just didn't stand a chance.
As for a democracy, I don't think that would work too well in a society where deities are not only common, but often physically present. A lot of monarchies drew their power from "divine authority", meaning people consented to monarchical rule because their God(s) told them to. The Royal Family has backing from the Goddesses, so thus the monarchy stands strong.