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.
Blockchain.info is the most secure, since the Blockchain.info doesn't hold your private keys. Well either way, it's best not to use a online wallet when you are handling large amounts of Bitcoins since it's not worth the risk.