The Shulhan Arukh (Yoreh De’ah 124:8) writes that if a "mumar" has contact with wine, the wine is prohibited to drink. In this context, a mumar is a Jew who has converted to another religion. What is the scope of this stringency?
The Rabbis teach us that one who violates Shabbat publically (mehalel Shabbat befarhesya) is considered to be a "mumar" (see Eruvin 67a), and is even compared to one who worships avoda zara (see Rashi, Bamidbar 15:41). Does that mean that wine which is touched by a mehalel Shabbat is prohibited?
Although in the Beit Yosef (Yoreh De’ah 119) he cites those who prohibit the wine, Maran does not bring this law in the Shulhan Arukh. Seemingly, the mehalel Shabbat himself is not literally worshipping avoda zara, and therefore this prohibition should not apply to him. Furthermore, one of the reasons for the decree prohibiting wine which was touch by non-Jews was to discourage intermarriage. It seems, however, that the prohibition of intermarriage does not apply to the Jew who is mehalel Shabbat.
There are Poskim (see, for example, Nekudot HaKesef 124), however, who write that wine touched by a mehalel Shabbat is prohibited. The Hatam Sofer (Yoreh De’ah 120) explains that although the wine of a mehalel Shabbat cannot be prohibited due to the fear of intermarriage, the Rabbis added a "kenas," a penalty, to those who flagrantly violate the Shabbat, including them in this law. R. Moshe Feinstein, in his Iggerot Moshe (Orah Haim 5:37), writes that this practice does not appear in the Talmud or the Shulhan Arukh, and is a minhag (custom).
Therefore, in practice, if we are hosting a non-religious Jew, we should be care that he should not touch our wine (i.e. wine which is not mevushal).
Some ask whether we may even give him wine to drink, as he should seemingly prohibit his own wine?! While some suggest that this is not considered to be lifnei iver (enabling a person to sin) since he can take the wine by himself, it may be simpler to explain that since the custom was enacted in order to discourage close, social contact, the person certainly cannot avoid close contact with himself! And therefore this prohibition would not apply to him.
Finally, we should note that this only applies to those mehalalie Shabbat who were raised with the awareness and knowledge of Shabbat, and chose to reject it. Furthermore, the Mishna Berura writes that one who violates the Shabbat, but is embarrassed in front of important people, is not considered to be a mehalel Shabbat befarhesya, and therefore his wine would be permitted.
