I thought fair to give the benefit of the doubt, and from what I've read - they complainant had gone to the FBI first, who had decided they couldn't investigate, likely due to time elapsed and lack of corroborating witnesses - and so I thought it fair that if the FBI themselves had decided not to investigate, that there shouldn't be political pressure to do so.
However, with more accusations emerging, some with corroborating witnesses, I think his nominations is becoming untenable. Again, I don't think the political parties should be directing the FBI etc. to investigate, but if they are choosing to investigate then that is their jurisdiction as such and will be interesting to see what they find.
The FBI investigation would be for his background file, not for a federal prosecutor to indict him. The White House definitely has the right to ask for such an investigation, and did so for Clarence Thomas.
I find it really weird that the two women who are making the accusations are not just okay with an investigation but are actively asking for it, but not Kavanaugh and his supporters in the GoP.
The GOP is happy to call this a smear job, but if it is, they should be the first to call for an FBI investigation. The FBI can determine if this is a coordinated smear job. Armed with that information, the Senate can proceed, or not. But instead, they want this to remain a he-said-they-said fight. It reeks of dishonesty.