Answer:
In any case, this type of view is just a short jump away from the notion that children do not have the intellectual capacity to make voting voices and we should recall that the same argument motivated the lack of enfranchisement of women and racial minorities (and in some countries racial majorities) for decades and centuries.
The second objection is that children are unduly influenced by others. Perhaps they are influenced by the media. Of course, on that score, it appears that adults are equally susceptible and the fact that children might watch different media may be a good thing for democracy. Perhaps teachers will have an undue influence. That may be worrisome but there are worse outcomes and, in any case, the political diversity of teachers is probably high enough that no single teacher could hold sway over large groups of children.
Certainly there is less risk of that than some celebrity holding sway over large groups of adults. Finally, perhaps parents will have too much influence. Again, wasn't that the reason why women were denied a vote -- on the claim that their husbands would have that influence when, more likely, it was the fear that they wouldn't that caused resistance to change.
The third objection is that very young children can't frame the issues or understand what the candidates are proposing. So if we push things to the limit, it is hard to imagine babies, toddlers or children who cannot read being able to physically vote. But this is no argument to wait until they are 18 (when they can drink as well as vote). This is an argument to wait until they are 8 or perhaps pass some basic civics test.
Consider the upside of enfranchising children. For starters, there would be engagement on a whole set of issues to do with them and also with families as a result of allowing children to vote. Now some have proposed that perhaps a child's vote can be held by their parents as a proxy until they are of age (see this discussion by Miles Corak on Demeny voting). It is true that this will bring family issues more attention but, of course, children may differ in their views on a number of issues from their parents.
But more importantly, by giving children the vote, they will be engaged early on and more interested in policy issues so as to formulate their own views. Democracy flourishes on engagement as much as it does on who gets to vote. Children may well be more likely to take this right seriously and also to take a longer-term perspective on many issues. That was certainly the case with my own children when I gave them a voice in my own voting.
When it comes down to it, if you are sceptical about all this, when you look into your heart as an adult, aren't you worried that by giving children the vote, that policies will change in a whole set of ways you don't want? That children won't share your views and that politicians will respond to that by acting in ways you don't want them to act. Perhaps you have an image of candy subsidies although you might want to check on that when you look at what happens with sugar in most countries!
And if that is really your objection then what you are saying is that you don't want a group to have the vote precisely because it will give them political power and reduce your own. And that is about as anti-democratic a view as is it possible to have.