Try first if you are able to write with sudo:
Code:
sudo touch /mnt/sda1/test
If that creates a file "test", then you can try this: Create a directory and set knoppix as it's owner. Then you can write to that directory normally:
Code:
sudo mkdir /mnt/sda1/data
sudo chown knoppix:knoppix /mnt/sda1/data
After that you should be able to write to /mnt/sda1/data normally.
ps. I don't know how to make it so that whole /mnt/sda1 can be written by a normal user, so this is what I normally use.