Older computers (e.g. with 32-bit CPUs) may require older versions of BOINC. These versions should still work; however, they may be unable to communicate with some projects (and with the BOINC server) because the "certificate authority (CA) bundle" is out of date. (This file is a list of servers used to authenticate the certificates used for HTTPS communication).

To fix this problem, get a new CA bundle from here, put it in your BOINC data directory, and restart BOINC.

On Windows, the BOINC data directory is normally C:\Program Files\BOINC\. This is a typically a protected directory and you won't be able to download the file directly into it. Instead, download it somewhere else, then copy it to the data directory. You'll get a dialog saying

You'll need to provide administrator permission to copy to this folder.

Click Continue or provide the main administrator's password.