Asteroid mining would disrupt the market for many rare elements, but I think it could be less chaotic than people think. It's not going to be like the Gold Rush where a deposit was found and prospectors flocked to it. It'll be a massive undertaking, almost certainly with multiple nations working together to achieve. The time between the project becoming public and the resulting mined resources being delivered would give markets time to settle on a price.
That said, yes, asteroids mining could certainly affect the price of gold, but not necessarily destroy all its value. There's a price floor set by the industrial applications for it.
With BitCoin, there is no floor. If no one wants to buy your BitCoins, you can't do anything else with them. They are just data representing proof of work, but no outside value.
Plus, in a world where you can mine asteroids for gold, suddenly there's potential for growth not only on Earth, but in the solar system at large, and there have to be industrial uses of gold on Mars...
Asteroid mining won't be able to significantly affect prices of gold downwell, definitely not until we bootstrap a real industrial presence in space - with the way we do space today, mission costs would eat up any profit you could make on shipping raw material down.
