Praise be to Allah.
It is permissible to write the Basmalah on invitations and other letters, because the Prophet (peace and blessings of Allaah be upon him) used to begin his letters by mentioning the name of Allaah. It is not permissible for anyone who receives a letter in which Allaah is mentioned or an aayah of the Qur’aan is quoted to throw it into the garbage or put it in any undesirable place. The same applies to newspapers and other similar papers: it is not permitted to mishandle them or throw them in the garbage, or to use them to wrap food or other things, because the name of Allaah is mentioned in them. The sin is on the person who does this, not on the one who writes it. And Allaah knows best.