The OSX InstallAnywhere version of Jalview is downloaded as a ZIP archive (install.zip) which is normally unpacked automatically to create an application called 'install' that you can run. If nothing happens when you double click the application, then there are a few things that could go wrong.
Installer doesn't launch because it was unpacked with a third-party archive tool
If you have chosen an alternative archive program for OSX to use to unpack ZIP files, then it probably didn't set the executable permissions for the application after it unpacked it. The easiest solution is to right or option click the install.zip that you downloaded, and choose 'Open with -> Archive Utility (10.X)' (where X is the numeric version of OSX that you are running); this will create a new 'install' application (probably called 'install 2') that you can now launch.
If using a different archiver doesn't work, or you are working from the console, then you need to delve into the installer's application directory. Open the console, cd to the directory where the install application is located (probably ~/Downloads), and type:
chmod u+x install.app/Contents/MacOS/install open install.app
Permissions problems result in an error shown in the OSX Console window that reads something like
([0x0-0x6a86a8].install) posix_spawn("/Users/jimp/install.app/Contents/MacOS/install", ...): Permission denied
If you see anything else, paste it into the search box on http://issues.jalview.org to see if it has been reported.
Related bugs in the Jalview Issue Database: JAL-746.