Inspecting the error log, the 500 error embedded in the downloader console is the result of a php timeout:

mod_fcgid: read data timeout in 45 seconds, referer: https://www.<omitted>
(110)Connection timed out: mod_fcgid: ap_pass_brigade failed in  handle_request_ipc function, referer: https://www.

According to Plesk Article ID: 127621, FcgidIOTimeout should be the same as PHP’s max_execution_time unless the latter is set to 0.

From Plesk 12.5 control panel for a domain use Apache & nginx Settings to add the following directive to both http and https, which increases the FcgidIOTimeout from 45 to 120 seconds :

<IfModule mod_fcgid.c>
    FcgidIOTimeout 120
</IfModule>

Simply OK this and re-run the Magento Downloader again.

Note: at the time of writing the Magneto Connect site is very slow, which may be the cause of this timeout.