Don't know about Coinbase specifically, but in general leaving your money with some small company in a jurisdiction you may or may not share is not a very safe idea. In fact, I'm pretty sure trusting third parties with their funds accounts for the majority of BTC lost by Bitcoin users.
I'm new to Bitcoin -- what's the alternative to Coinbase? I've read you can store your wallet on your computer, but I would be afraid to do that because then I'd lose everything if my computer crashed or was stolen, right?
I need to read more on how Bitcoin works..