No, you do not need to have the client running to receive payment on an existing address. Your transaction would go through because bitcoin transactions aren't contained on a local file, but a distributed ledger called the "blockchain."
You don't need to be logged in or anything to receive bitcoin. What's important is to keep your wallet file containing your bitcoin private keys secret and backed up, as without those, you have no access to the address with your bitcoin.
In fact, a lot of people, fearing their computers being compromised by hackers, actually advocate you store large amounts of bitcoin in a wallet that's generated offline and stored on an offline computer, USB flash drive, or piece of paper. This practice is called cold storage, and is a very good idea for large amounts of bitcoin you don't want to lose and have no intention to spend quickly. So, the paranoid strategy would be to never have your wallet online, believe it or not.
A friend of mine received Bitcoins for a transaction last year, and they were sent to him in a file. He is having a hard to locating the file. Mind you he did not have a wallet at the time. Does anyone know what that file might look like, or how to go about locating it? Thank you, Matt – None – 2013-12-03T04:02:13.937
1please post this as a new question, don't post questions as answers. See [ask] for more information. – Murch – 2013-12-03T09:56:07.363