You can see free space for all mounted filesystems with "df" command (use in Konsole) or "KDiskFree" command from KMenu.

In Knoppix 4.0.2 CD KDiskFree is at KMenu->System->More Applications->KDiskFree
You can also press "Alt-F2", then write "KDiskFree" and press Enter.

Those tools can show quite many devices and free space for them. Look for a device with "knoppix.img" in it.
e.g. for me the device for persistant home is /mnt/hdb2/knoppix.img

Example output from df:
Code:
knoppix@1[knoppix]$ df
Filesystem           1K-blocks      Used Available Use% Mounted on
/dev/root                 2471        22      2449   1% /
/dev/hdb2              5313432   4919712    393720  93% /cdrom2.loop
/ramdisk                219712     11036    208676   6% /ramdisk
/UNIONFS              14797436  11475880   3321556  78% /UNIONFS
/cdrom2.loop/CD.iso     711124    711124         0 100% /cdrom/
/dev/cloop             1941958   1941958         0 100% /KNOPPIX
/UNIONFS/dev/hdb2      5313432   4919712    393720  93% /mnt/hdb2
/mnt/hdb2/knoppix.img
                         99150     85037     14113  86% /KNOPPIX.IMG
/UNIONFS/dev/mapper/LVM-data
                      12635768   9522888   3112880  76% /data
So this tells that I have used 86% of the space available in persistant home.
And I have 14113 kB space left.

If you want to know what is taking all that space, "filelight" is good for that.
Press Alt-F2, write "filelight" to command-box and press Enter.
Then select Scan->"Scan Home Directory"
(filelight is not in 4.0.2CD but is in 4.0.2DVD - It's probably also in 5.0.1DVD)