-Obviously, public opinion differs in different monarchies, but living in one constitutional monarchy - Norway - I feel that precisely as the monarchy is gutted of real power, its resilience is strengthened.
One could quip that we need someone to inaugurate new infrastructure works, provide soothing, apolitical speeches in time of need and serve as an unelected symbol of our country - the 'unelected' bit being important, as otherwise we'd just end up with some has-been politician put out to pasture, someone probably the majority of the population had never voted for and don't see as 'their' head of state.
Now, on principle, the monarchy is laughable, anachronistic and undemocratic - but donning my pragmatist glasses, I see it as probably the best way to ensure we have a (symbolic) head of state for performing just those duties where you need someone not overtly political. As long as the royals feel like doing the job, I say let them.