We're pretty certain our star is stable--we do live on a planet that has had hundreds of millions of years of life on it--but it wouldn't have to kick up much of a flare or something to make life difficult on Earth.

Besides, it's a good idea from an evolutionary standpoint to spread the species around.

I don't think we're going to come up with warp drives any time soon, but there are other options like generational starships, or possibly relativistic-speed starships.

The generational starships might be a natural outgrowth of solar system space colonization, particularly of asteroids. Get a few generations who have grown up on a huge nuclear-powered asteroid colony, and at some point you will find that it doesn't matter to their lifestyles whether they and their children are orbiting the sun or somewhere between here and Alpha Centauri.