Judaism has mostly not been proselytizing. By the time of the Talmud it is clear that converts were to be dissuaded, not recruited. That's been the general trend for the past 1500 years or so.

However, the Talmud represents a particular sort of Judaism that survived the Great Jewish Revolt, and had been affected by the huge upheaval of it. There are a couple pieces of evidence that this was not always the case.

First, Josephus records the story of two powerful non-Jews, Adiabene and her son Izates, converting to Judaism. It includes two different Jews doing a degree of convincing to convert. Exactly what isn't clear. But one "taught them to worship God according to the Jewish religion", and the other convinced Izates to be circumcised after he'd already begun following some sort of Jewish practice. While this is interesting mostly for the ancient view of converting to Judaism, it also indicates that there may've been some proselytizing. Maybe not missionizing, but certainly being willing to educate others in Judaism in a way not dissuasive to their joining it.

Secondly, Josephus records that the Idumeans (aka the Edomites) were conquered by the Hasmoneans, were actually forcibly converted. It's a bit more complicated than that, because it can be debated to what extent it was forcible, and to what extent it was an intentional integration into being Jewish. But it is something. This, however, seems to have not been terribly popular among all Jewish sects, who may've already seen conversion as something to, if not discourage, definitely not force.