When running from the live CD:

When Knoppix starts up KDE, you'll see icons for all of your disk partitions on the left-hand side of your screen. Simply right-click on one of them to bring up a context menu. Choose "mount" to mount the partition. Then choose "Actions > Change read/write mode" and a dialog box will come up asking if you really want to make the partition writeable. Click "yes".

Warning: Do not try to write to an NTFS partition! Data corruption may result.

(Can't find this question answered in a FAQ; it probably should be in one.)