They are a bay area Y combinator startup and also funded by FundersClub among other angel investors. They just also received $25M from Andreessen Horowitz which is a top VC firm and super smart guys. Going through this investment process alone makes them in my book as about legit as you can get.
I've heard of people having problems with coinbase, but it seems mostly due their risk analysis system where they don't allow certain users to purchase bitcoins perhaps after a request has been made. I don't know about this personally as it never happened to me.
I've purchased over 100 BTC from coinbase. Never a problem. This includes instant purchases as well as purchases that takes many days to "clear".
As with any startup, there will be growing pains, and bugs to work out both with the software as well as customer service issues and policies, but as far as legitimate honest business professionals, I trust them.
Disclaimer: I have no financial interest tied to Coinbase (wish I did actually)
I agree with this guy - I have done many dealings with coinbase and never had an issue.