This is mostly a compilation of various things that have worked to resolve some Mac issues with the Launcher during beta. They are not identified as resolving the existing Launcher issues but I though it was worth putting them together for those that want to try them.

Please post how it goes especially if any of them do help.

Do these one at a time not all together to test.

Repair the ESO Install

In the Game Options drop down menu (next to the install in left window) of the Launcher select Repair. It may not show if the install has not been completed previously. Click and let it Verify the installed files. Usually it will take some time to verify each downloaded section then download any missing files and Complete.

Redownload the Mac Installer and install Launcher

Move the existing Launcher from the Zenimax Online folder in Applications (to Desktop)

Then go to your ESO account page and redownload the Installer and follow instructions. Normally it will auto launch the Launcher.

Try again.

Rebooting Mac & Modem/ Router

Shut down your Mac, reboot your modem/ router then boot up and try again.

Switch off OSX Firewall

Disable the OS X Firewall during download

System Preferences > Secuirty & Privacy > Firewall

This is in OS X 10.9 Mavericks, It may be slightly different in other versions.

Change DNS

On the Mac, to change the DNS server IP addresses go to:

System Preferences > Network > Advanced… > DNS

Note the existing ones usually automatically added by your ISP then change them for the Google ones of other you prefer. Remember to Apply changes after clicking OK when done.

Google: 8.8.8.8, 8.8.4.4

Renewing the DCHP Lease.

System Preferences > Network > Select the network you are connected to (indicated by green circle)> Click Advanced > Click Renew DHCP Lease

Switching off Auto Proxy Discovery

-Navigate to the System Preferences under the Apple Menu.
-Open up Network in System Preferences. (Usually located in the third row, third icon.)
-Make sure the network connection with a green dot next to it is selected.
-Click the Advanced button in the lower right corner
-From the drop down menu, click on Proxies.
-In the Select a protocol to configure list
-Uncheck Auto Proxy Discovery

Afterwards to need to setting them back as before. This is just for testing. Leave the Firewall disabled whenever trying to download or update. All these things have helped various Mac connection issues but will not necessarily resolve this issue. But worth trying.

If possible disable the firewall in your router while testing but enable again once finished.