When did the Jewish people stop believing in Jesus as the Messiah?

Answer 1

The question is based on the incorrect supposition that the Jews of today had ancestors who accepted Jesus's claim to "Messiah-hood" and later rejected it. This is not true. Jews consistently disbelieved Jesus's claims because they contradicted Torah values and Jewish customs. It was only those Jews that Jesus convinced to believe in him that became Christians, the rest remained as they were, as if Jesus had never lived.

Answer 2

The didn't "stop" because they never started. According to our tradition, the vast majority of the Jews at the time didn't hear of him. The Torah-sages (Rabbi Yochanan ben Zakkai, Rabbi Yonatan ben Uziel, Chanina ben Dosa, Bava ben Buta, Shimon ben Hillel, Rabbi Eliezer, Rabbi Yehoshua, Rabbi Akiva, and hundreds of others) were active at that time and their yeshivot (Torah-academies) were flourishing. Their tens of thousands of disciples and hundreds of thousands of sympathizers were active in the Jewish world in that generation; they were the leaders and the forefront of Judaism. As Josephus (Antiquities book 18) writes, "the cities give great attestations to them." The great majority of Jews loved their sages and their Torah.
The unlearned class of the Amei-haaretz (ignoramuses) was a small fringe of society, but even they would and did lay down their lives in order not to violate anything of the Torah. As one ancient historian famously wrote:
Hecateus declares again, "what regard we [Jews] have for our laws; and we resolve to endure anything rather than transgress them." And he adds: "They [Jews] may be stripped on this account, and have torments inflicted upon them, and be brought to the most terrible kinds of death, but they meet these tortures after an extraordinary manner, beyond all other people, and will not renounce the religion of their forefathers."

No one (even any who did hear of Jesus) - would have given heed to what was and is consideredunacceptable for us. The few who came in contact with him soon lost interest, and the early Christians felt the need to turn to non-Jewish centers of population in order to gain adherents, while the Jews remained Jews.

Rather, you might prefer to ask "What does Judaism notbelieve about Jesus." And the answer is that we do not believe that he is or was anything other than a regular human being.

(See: What do Jews believe God is like?)

We may also note that according to our tradition, prophecy ceased about 340 years before the birth of Jesus; and public miracles stopped even earlier.

Here is a related topic:
The word "messiah" is the transliterated form of the Hebrew "moshiach." The word moshiach means "anointed." The title of moshiach was given to any person who was appropriately anointed with oil as part of their initiation to their service of God. We have had a number of meshichim (plural) in the form of kings and priests. There need be nothing supernatural about a moshiach.
This being said, there is a prophecy of a future moshiach. However, this is a relatively minor topic in Judaism and the Tanakh.
The Jewish requirements of the messiah are:
* Build the Third Temple (Ezekiel 37:26-28).
* Gather all Jews back to the Land of Israel (Isaiah 43:5-6).
* Usher in an era of world peace, and end all hatred and oppression. As it says: "Nation shall not lift up sword against nation, nor shall they learn war any more" (Isaiah 2:4).
* Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: "God will be King over all the world. On that day, God will be One and His Name will be One" (Zechariah 14:9).
* The messiah must be descended on his father's side from King David (Genesis 49:10 and Isaiah 11:1).
* The messiah will lead the Jewish people to full Torah-observance. The Torah states that all of its mitzvot (commands) remain binding forever.