Yes you can, when you insert the USB stick and click the icon in the desktop Knoppix automatically mounts the drive, but it mounts it read-only. To change this to read-write, right click the usb sticks icon on the desktop, go to Actions and the first in the list should say something like "Change to read write mode".
Alternatively instead of letting knoppix mount it read-only in the first place just issue this command in the shell:
sudo mount -o rw /dev/sda1 /mnt/sda1
Or sdb1 in your case.