Greater public support strongly increases the probability that a senator will vote to approve a nominee, even after controlling for standard predictors of roll call voting. We also find that the impact of opinion varies with context: it has a greater effect on opposition party senators, on ideologically opposed senators, and for generally weak nominees.
More discussion, and some pretty graphs, below.
Kastellec, Lax, and Phillips use the method of multilevel regression and poststratification (Mister P, as we call it) to estimate public opinion by state, which is important for their goal of connecting state opinion to Senators' votes. Here is some of what they find:
Opinion about some of the justices varied a lot by state; see here:
And here are their graphs showing the connection between state opinion and senators' votes:
As they discuss in the article and show with a regression analysis, the correlation between state opinion and senators' votes remains, even after adjusting for senators' liberal/conservativeness. You can read the full article for more details.