The conventional wisdom is that this effect, to the extent it exists, will help the GOP more in the Senate than the House. Nate Silver, who has the most detailed and careful assessment of this effect, mostly agrees with this:
"From a 35,000-foot view, the story in the generic ballot numbers is largely one of stability.1 If you want to be more precise, however, the trend in the generic ballot now depends on what point in time you’re comparing against. The GOP’s current deficit on the generic ballot, 8.0 percentage points, is a bit worse than it was before Kavanaugh was nominated, when it was 7.4 percentage points. It’s slightly better than it was when Ford’s name was disclosed, however, when it was 9.1 percentage points, or since just before last week’s hearings, when it was 8.6 percentage points.
Trump’s approval ratings have largely followed the same trajectory as the generic ballot, having slumped in early-to-mid September and since rebounded slightly. It’s not clear how much of that is Kavanaugh-related, however, as the president was dealing with a lot of other news in August and early September, such as the guilty pleas of Paul Manafort and Michael Cohen. Merely staying out of the headlines while Kavanaugh was the lead story may have helped Trump’s numbers revert to the mean. It’s also not clear if Trump’s numbers have improved since last week’s Senate hearings; in the all-polls version of our average, they’ve gotten a bit better, but in the registered-voter version, they’ve gotten slightly worse. (There hasn’t yet been time for the polls to reflect any impact of Trump having mocked Ford at a rally on Tuesday night.)...
In the Classic and Deluxe versions of our House forecast, Republicans’ numbers have reverted back to where they were in early September, with around a 25 percent chance (1 in 4) of keeping the House. However, they’re somewhat better than than they were in mid-September, when their chances had slumped to as low as 17 percent (about 1 in 6) in the Classic version of our model. They’re also a bit better than before last week’s hearings, when they were around 20 percent (1 in 5)....
Republicans have been favored to keep the Senate all along. But their position has improved quite a bit over the last week in all three versions of our model. In our Classic Senate forecast, for example, Republicans are now 77 percent favorites to hold the chamber, up from 68 percent before last week’s hearings.
A lot of this comes down to Heitkamp and North Dakota, where Republican Kevin Cramer is now a 2-to-1 favorite despite the traditionally strong performance of opposition-party incumbents in potential wave elections....
Republicans are generally doing worse in district-level polls than you’d expect them to do in generic ballot polls, even though district polls are almost always conducted among likely voters. One possibility is that Kavanaugh is helping with Republican base turnout, but also hurting the GOP among swing voters with a high propensity to turn out, such as suburban women.
Overall, I’m inclined to conclude there’s actually something there for Republicans — that their position has genuinely improved from where it was a week ago (although, not necessarily as compared to where it was a month ago)."
That all sounds pretty reasonable to me. There's some improvement for the GOP, but not enough to really change things for the House, while the Senate, always difficult, has become even more so for the Democrats. I also would not be surprised if, once Kavanaugh is confirmed--and I think it's quite likely at this point--positive poll effects for the GOP somewhat dissipate, as their base's fury subsides while Democrats are mad as hornets he got confirmed. We shall see.
To further illustrate how things on the ground are probably not changing all that much, it's worth checking out the latest Sabato Crystal Ball ratings changes. The topline in their detailed report is this:
* 11 House ratings changes, all in favor of Democrats.
* Five gubernatorial ratings changes go in different directions but are generally better for Democrats.
* Only one change in the Senate as the battle for that chamber remains in something of a stasis.