I have a Nikon D3000. I want to protect every image that is taken with the camera, but I would like to make this automatically. I mean every time a photo is taken the camera should protect it itself. Is this possible?
No, there's no such option on D3000.
If you're worried about photos getting deleted by yourself or somebody else, you could try getting your hands on a non-rewritable SD card, like Sandisk WORM or Toshiba Write-Once. Note that you can't reuse those cards after they get full, and you could still lose your pictures if somebody has access to your camera and removes the card.
Another, perhaps more realistic, option for providing protection to your photos is using an Eye-Fi card to instantly upload the photos to network. For this to work, there must be a Wi-Fi network available and accessible at your shooting location.
I've had a look in the manual (http://www.nikonusa.com/pdf/manuals/noprint/D3000_ENnoprint.pdf (page 100), and it makes no mention of making this the default mode. Sorry...