Good question. Strictly speaking, we don't know how commonly held the belief was (as we do not have a huge number of documents from that exact time period, and certainly "opinion surveys" didn't exist then). However, the IVP Commentary does a good job of explaining the probable background based on rabbinic comments from the following centuries. I will summarize it here:
- It was a common belief that all disease and disabilities arose from sin. This can be seen, for instance, in the rabbinic saying "there is no death without sin and there is no suffering without iniquity" (b. shabbat 55a).
- This belief causes a problem for people born with disabilities with two possible solutions:
- 1) their parents' sin was responsible. This position can be justified by for example, Exodus 20:5 and Ezekiel 18:20.
- 2) the person had someone sinner before birth. Whether this was possible or not was a matter of rabbinic debate. Some rabbis thought it possible, while others thought it impossible. Its not clear how widely held the prenatal sin view was, but apparently it was widely held enough to generate debate at least.
- The disciples were thus probably looking for an answer to 2) when they asked the question.
As to how someone could hold the view of sinning in the womb, in Genesis Rabbah 63:6, Rabbi Yonhanan commenting on the story of Jacob and Esah, interprets the words "struggled together" in the womb and "this one ran to kill this one and this one ran to kill this one". In other words, that the babies tried to kill each other in the womb.