In and of themselves, birthstones are shiny rocks created by God. As such, they are, like all creation, "very good" (cf. Gen. 1:31). Properly speaking, the devil cannot create anything, and so natural objects cannot be said to be "satanic." That said, the Church does warn against superstition and sorcery. The Catechism of the Catholic Church states:
Superstition is the deviation of religious feeling and of the practices this feeling imposes. It can even affect the worship we offer the true God, e.g., when one attributes an importance in some way magical to certain practices otherwise lawful or necessary. To attribute the efficacy of prayers or of sacramental signs to their mere external performance, apart from the interior dispositions that they demand, is to fall into superstition (CCC 2111).
All practices of magic or sorcery, by which one attempts to tame occult powers, so as to place them at one's service and have a supernatural power over others -- even if this were for the sake of restoring their health -- are gravely contrary to the virtue of religion. These practices are even more to be condemned when accompanied by the intention of harming someone, or when they have recourse to the intervention of demons. Wearing charms is also reprehensible. Spiritism often implies divination or magical practices; the Church for her part warns the faithful against it. Recourse to so-called traditional cures does not justify either the invocation of evil powers or the exploitation of another's credulity (CCC 2117).
For the most part, the practice of assigning various stones to the months of the year is both a pretty symbolism and a marketing stratagem by jewelers (source). Insofar as that is the case then, birthstones are neither superstitious nor magical. Only in those cases when someone puts undue faith in a stone (superstition) or uses the stone as a charm or divination tool (sorcery), would there be a moral issue. The issue though would be with the false belief and/or forbidden practice and not with the stone itself.