Is there a source for the claim that being a Kvater (person who carries a baby to/from his Bris) helps cure infertility?
The author of the Sefer Shaleiach Teshalach (English edition p. 98) writes that he was personally told by R' Elyashiv and R' Chaim Kanievsky that "there is no Midrashic or Rabbinic source whatsoever connecting the honor of kvatter with being blessed with children." He also cites R' Shlomo Zalman Aurbach to this effect (from the Sefer V'aleihu Lo Yibol). [Link here - search term "kvatter".]
I once tried doing a computer search for this, and the earliest sefer that mentions anything about it was by a contemporary author (written in the 70's, IIRC).
In the Book Bris Secrets a source is brought from the Sefer Sharie Ahron a Posuk in The Torah!!! it says:
וְאֶתְּנָה בְרִיתִי, בֵּינִי וּבֵינֶךָ; וְאַרְבֶּה אוֹתְךָ, בִּמְאֹד מְאֹד
Translated in This context as:
Ones who brings the Child into My Bris (The Kvater) that is between me and you and I will Multiply you very Much.
There are many sources against such an idea. To many Rishonim such as the Rambam the idea of some action or vehicle of Mitzvah to bring about some "magical" outcome would be extremely problematic. Even to suggest that by the performance of a Mitzvah you raise your level to merit a new Providential relationship to receive the desired outcome would be a stretch when applied to merely having some honor in assisting to do the Milah. However, if the person believes that by being a Kvater it may help with their infertility the psychological impact might make a difference. There have been many studies that have shown the relationship between the mind and the affects on the body. Certainly infertility at times may be impacted by the psyche of the parties involved.
So you may not want to stop the practice, but I am not sure you want to encourage it.
Lephi Aniyas Daati