Looking at the screenshot posted in the question, it seems that Magento connect credentials are not required here.
This pop-up seems to be applied by .htpasswd file.
You need to check .htaccess file in the document root and try to find the location of .htpasswd.
Now, you either need to remove below code to remove this authentication:
AuthName "Restricted Content"
AuthUserFile /[location of .htpasswd file]/.htpasswd
or, you need to navigate to [location of .htpasswd file] and change the password in following format:
Below is an example:
You can generate the encrypted password from here.
Update: After either removing the pop-up or logging in from the pop-up, you need to login to Magento Connect using your own administrator account credentials.
Please let me know if you have any other question.