Keeps asking Connection Information upon update to 3.5.1 (but not for plugins?)

WordPress primary choice is to do upgrade using direct file IO. So if it’s asking you for FTP credentials it means that it has no access to these files.

The easiest way to fix this will be changing file permissions so WordPress could modify them. Of course if you changed those permissions to improve security it’s not a solution. (Well, you still can modify these permissions, run upgrade, and then set them back).

If you provide correct FTP credentials when asked and the upgrade still doesn’t succeed, it means that WordPress is unable to connect using given credentials. Use this link to read more on using FTP to upgrade (http://codex.wordpress.org/Editing_wp-config.php#WordPress_Upgrade_Constants).