Would I be sinful if I were to give my old computer to an employee who does not care about the presence of God's name on the papers he throws in the wastebasket, knowing this beforehand?

One aspect of revering the rituals of Allah is protecting anything upon which His exalted name is written from desecration. Therefore, if you give a computer to someone and you know that they will use it to print papers containing the names of Allah Almighty and texts from divine revelation, and then discard them, you must repent for enabling them to do so, because that is aiding in sin. If they do not commit to respecting these (papers), then take back the device if you can. Otherwise, you must advise them and explain the obligation to respect the names of Allah and the texts of divine revelation. If you fulfill your duty of advising and enjoining what is right, then you have discharged your responsibility.