Some of the installed files under Resources/ don't give world read and execute permission. Admin users (in the admin group) can play the game just fine, but I found that I needed to manually fix permissions for the game to be playable by non-admin users.

In Terminal in the directory

/Applications/Aqua Mines.app/Contents

I ran the command (as an admin user)

sudo chmod -R o+rX .

(note that the X is followed by a space and a dot)

to fix up the permissions and everything now works.