If it was working before you upgraded, then the problem is that you forgot to install the "CA cert bundle" that SSL (and therefore curl, which is used by Transmission) needs to verify the server.
The other case is when the server doesn't have a valid certificate, you could ignore the invalid/false/whatever certificate (at your own risk) with some parameter in an environment variable which I don't remember right now.
Correction: By default Transmission doesn't verify the server. The environment variables are user to enable that verification.
TR_CURL_SSL_VERIFY
CURL_CA_BUNDLE
Those are the environment variables.
Perhaps the upgrade has enabled that option... in other words, the person(s) who made that upgrade modified the source code or set the first variable in the start-up script. Either that or libcurl made the change (perhaps a bug since Transmission sets the option not to verify).
What are the values for the environment variables. I tried setting them but to no avail. I'm having the exact issue above as well, with Ubuntu and the newer versions.
Edit:
update
I uninstalled LAMP and removed everything, then re-installed transmission daemon and the deb package and still no go. Any Ideas on how to fix this? I'm on Ubuntu 12.04.