I have been having this problem for the last 2 months and opening a case at LMI does not help- they have been useless: Both the LMI client and Safari are unable to connect to LMI. For example, when I try to log into this support site I get the message "Can't establish a secure connection to the server "secure.logmein.com." When I try to open the client I get "Connect Failed" and I'm unable to make a connection to the LMI server- no list of computers, etc.
I CAN connect via Firefox (which I'm doing now) and so my only way of connecting to remote computers is via the web interface there. I assume that Safari is using the built-in certificate store while Firefox uses its own.
I have completely uninstalled anything I could find relating to LMI. I have gone into the certificate store and told Mac OS to trust the LMI certificate, but no luck there. I don't want to reinstall LMI until I have Safari working, because they seem to have the same problem.
I need suppoprt from someone who actually knows what they're doing. Help!