For the most part, the reason why I think that democracy matters is that people should have a say in the societal structures that they live under. I came close to this idea in Flag Change: Who Should Vote?
If you have to live with something and a reasonable person would consider that you have the awareness and knowledge to make a reasonable decision in the course of that something, you should have a say. This is, basically, the principle underlying democracy... rule by the people (reign by the people is somewhat different mind).One notes that this appears to be advocating qualified suffrage when removed from its context. I do not favour qualified suffrage except on the basis of age: everyone over eighteen should be allowed to vote. In fact, I personally favour everyone over the age of sixteen or, at the very least, a tethered age. Naturally, that was explained in the aforementioned post:
This is why some people have proposed a tethered voting age... you can vote if you turn 18 within x period of time after the date of the election (18 is also variable, of course). There is also the issue that if we have a mandatory schooling age of 16, we are pretty much saying that we expect people to be doing adult things and having adult lives from an age two years younger than that of the voting age.Adultness is also something to be wary of. The point being made is not that independence (i.e. having flown the nest: no longer being strictly a dependent) should determine that one can vote but rather participation. By the nature of our society, we assume that persons over the age of 18 participate. In fact, for almost all intents and purposes it is 17 because one is up in front of adult courts at 17. The problem identified above is that the way we have structured our school system, as a society, means that we acknowledge that people participate at an age younger than 18.
If you are wondering, this is why the age of eighteen is not so arbitrary as it first appears. While I am at a loss to explain why society chooses eighteen, of all possible ages, the point is that it understands eighteen as this moment of participation. It's still not completely non-arbitrary (e.g. plenty of fifteen year olds participate in political/philosophical discussion) but there is some non-whim based line of reasoning in there.
Which brings us back to the point quite nicely. I've just explained that we ought to vote (er, have democracy) because we should have a say in the societal structures we live under/with. I don't really know why that should be so. After all, while this is something a bit like the right to self-determination I don't know the philosophy underpinning (and, I daresay, nor do most people), provided you don't count "it seems right".
I could say that democracy is a guarantor of rights more generally. The issue with this is twofold. Firstly, why do those rights matter? (It gets a bit recursive for me because my major response is: people want them.) Secondly, is democracy really a guarantor of rights? It seems to me that this is not necessarily so. After all, it's very easy to have rights. It's a lot more difficult to have access to those rights: to actually be able to utilise them. Democracy follows from a situation where people can enjoy their theoretical rights (even if these are few in number), not the other way around. Democracy is, though, definitely a good way of preserving rights in a stable, non-corrupt democracy (and this is not a trivial thing to say).
On the other hand, perhaps it is sufficient that people want to have democracy. That is, the philosophy of utilitarianism is a justification. Basically, I am saying that when democracy doesn't harm anyone, if people want it, then the way to maximise benefits is to give people democracy. This is pretty easy to apply in practice. Why? Well, basically democracy is only harmful in specific circumstances.
Strange as it may seem, it's not a good idea to try and adopt what is inherently divisive if continuity and stability do not exist. If you do try this, then the inevitable outcome is that the divisions created by voting provide more destabilising momentum to the structure. Not good. Also, don't try to impose democracy: democracy is about groundswell, if people don't really accept it then there is a good chance it provides further destabilisations. Impositions can work with continuity (see: Japan after WWII as an example) but it is the continuity that matters, not the democracy. In practice, democracy requires a strong state structure that survives different leaders otherwise it falls down. It also requires trust -- trust that people will do what they are meant to -- in the sense that corruption can destabilise the stable (which is why strong democracies are also very transparent: but these aren't independent concepts).
However, I think, given how few people seriously reject democracy, it is not sufficient to move outside the democratic paradigm. That is, for all intents and purposes, the reason why democracy is good, why we should care when veneers are paraded as the real deal, is that the people ought to rule themselves. That is, people should have a meaningful say in the societal structures they live in. And yes, that means not just citizens should have the right to vote: permanent residents too.