Actually, I take it back, in my last transaction, I didn't pay a fee which I saw that as weird.
Maybe they do pay the fee for their customers.
I think of Coinbase as one of the more secure online wallets for btc.
Ya. 2 things I want to say. First, they only pay the fees for transactions over 0.001. Second, it is as secure as you want it to be. If you want you can enable two factor and make sure you phone never goes out of sight. You can make sure that "remember me" isnt checked, etc. The safest you can get is paperwallets.