Absolutely continue. Innocent until proven guilty. No evidence that he's guilty(thus far). Vote should happen by mid next week as it passed the committee earlier today to send it to the floor and the 7th FBI check will be pretty quick.
I feel like I can be of help on this notion. For those that don't have a legal background, there is actually no explicit guarantee of "innocent until proven guilty" established in the US Constitution. Instead case law and precedent from the Supreme Court have established the notion as a combined part of the 5th, 6th, and 14th Amendment. There is no dispute in American legal precedent that there is presumption of innocence.
However, this doesn't mean the idea isn't misinterpreted. Presumption of innocence is ONLY a guarantee for criminal defendants. To better place that in the wording of the Constitution, such manners of due process are only guaranteed when life, liberty, or property are at stake. Finally, it must be a government actor that seeks to seize life, liberty, or property.
In Judge Kavanaugh's case, he is not a criminal defendant nor is a government actor seeking to seize his life, liberty, or property. Judge Kavanaugh is not owed this presumption. Frankly, as a Judge, he knows in this setting he isn't owed much of anything. However, Senate committee rules due try to provide some forms of due process to have a fair hearing when questioning nominees.
For a lot of good reasons which I wont' get into here, presumption of innocence has little worth outside of the criminal context. Civil cases do not have this presumption for instance. Really it is meant to be a check on the executive branch to prevent unfair trials and proceedings where the government already enjoys more power.