Origin seems to cause a lot of difficulties. For all those who are stuck with their installer, here is some information.
Firstly, if you have made any attempts at all to install the game, look through this guide and get rid of any of the stray files mentioned. Even if your game has not properly installed, it is possible that some files are there, so check and remove them so that you have a fresh start.
http://bluebellflora.com/the-sims-3-on-a-mac/how-to-do-a-clean-uninstall-and-reinstall-on-a-mac/
Remove the Origin client as well and start from the beginning. Be sure that you have a stable internet connection. Check is that your Mac is sufficiently up-to-date to be able to run the Origin client. It requires at least OS X 10.6.8 with an Intel Core 2 Duo processor or better.
To download and install games through Origin:
First install the Origin client for Mac and make sure it is up-to-date. The latest version can be downloaded from here: http://www.dm.origin.com/mac/download
Additional advice on how to download games (thank you Tremayne for this tip):
When downloading from Origin, do NOT use the first install icon. Move your cursor over the game cover and click on the information (letter i) and then click on the install button from there. For some reason, downloads from Origin frequently become corrupted. Using the i reduces the chance of your download being corrupted. If you have already been unfortunate and have a corrupted download, then it is important to remove all traces of this before trying again, by doing a thorough, clean uninstallation first.
Are you running anti-virus software or any clean-up applications? If so, disable these, as they might be deleting files that Origin needs!
Install the base game and patch it up to date. At this point, launch the game and check it works. Then install any expansions in order of release date.
Caspin