Is there any way to run an older version of the Steam client?

Yes. But it's not really recommended.

There are sites out there like FileHippo or OldVersion.com that will let you download the older versions of the Steam client. However, it's REALLY not recommended to do so.

With that being said, unless you want to block Steam from checking its servers for a new update, this will almost always involve the usage of a pirated Steam client.

Usage of a pirated Steam client is against the Steam user agreement and cause problems for you as the user, the least of which is the permanent banning of your Steam Account as well as being banned from any games which uses the Steam VAC.

If your OSX Client isn't working, before using an older version, try heading to the Steam community forum and see if there are others with the same issue. If possible, reinstall the client. (You SHOULD be able to do this without having to redownload the games already installed.)

Use Paul's advice to get an old version (non-pirated). Personally, I got the last working version off my backup drive. Then I googled for advice on blocking updates. It's actually quite easy.

Make a text file containing 1 line: BootStrapperInhibitAll=Enable

Name it Steam.cfg. Open the Steam app with right-click -> Show Package Contents. Put your Steam.cfg file inside Contents/MacOS within the app. Now it won't auto-update.