Ah, but the ones that would be influenced simply by curiosity and wonder typically don't need much convincing. I work at that angle (curiosity and wonder) with the little Oni, who then passes it on to her friends - I'm such a corrupting influence.
If I ever have kids, I'm going to indoctrinate the hell out of them.
But I think a lot of people just don't know how amazing the universe is, because they haven't been exposed to it. It's a large part of the reason why I love the Symphony of Science project. It's a way of exposing people to amazing concepts in a way that's easy to grasp. Now, that only indirectly ( or partially ) affects the exploration of space, but then, all science is connected, so I don't think it really makes sense to talk about the amazing things we could discover on the planets in the solar system without getting into things like biology, chemistry, and the physics of stars and other things.
One thing that, even as I write this, is becoming very clear, is that the erosion of science education that people like Carl Sagan feared ( and that we see dramatic examples of all over the world ), affects a lot more than just peoples' career choices.
There is of course one practical aspect of space exploration that makes it pretty much an imperative: one day, the sun is gonna go red giant and consume our little planet.
Sure, it's a long way off yet - but I figure, why not start planning early?
We're fairly overdue for a catastrophic asteroid strike, too.
Yeah, there's a chance Apophis will hit us in the not too distant future. A slim chance, but a chance all the same. Its name is not incidental, though maybe a touch dramatic. You're right about the sun, too, of course. I looked it up, and a quick google search indicated that our sun would turn into a red giant in about 5 billion years. But don't worry. If you didn't know, and need some cheering up, you should know that the Andromeda galaxy is on a collision course with the Milky Way, and they're predicted to collide in 3 - 5 billion years. In other words, all life here will probably be annihilated before our sun consumes us. Isn't the universe just full of wonder? It's like Christopher Hitchens used to say, "A whole lot of nothing is coming right toward us."
If that's still too cheerful, let me leave you all with this quote, allegedly by Kurt Vonnegut ( I first heard it quoted by Lawrence M. Krauss, another amazing individual ). Now, he said "Things are going to get unimaginably worse, and they are never, ever, going to get better again!"
Such is the nature of the universe we live in. And I don't know about you, but I find it unbelievably fascinating.