“When I hear his steps outside my door I lie down on my bed, open my legs and think of the Donald."
In a word, no. The Democratic Party has a good chance of taking a majority in the House with the current status quo, and opposing Kavanaugh, whose rulings and positions pretty much fall along partisan lines, will not change that. The Democrats have a negligible chance of winning control the Senate where they would would have to retain ten out of twenty-six seats (including two independent candidates in caucus with the DNC) up for contention in ostensibly pro-Trump states and gain two more in unfavorable races against the nine Republican-held seats, seven of which are incumbents, which would be a historical upset. This isn’t wholly unprecidented—it’s the same numerically as happened in 2008–but holding that number of seats up for election hasn’t been shown since 1982, and in the post-WWII era there has never been that combination of holds and gains by Democrats. The Democrats actually be statistically fortunate not to lose a seat or two in this scenario.
Regardless, with a Supreme Court appointment for life, there is good reason to take a principled stance, even if it is a losing proposition in the short term. And the supposed strength of the anti-abortion contingent as single issue voters with a chockhold on swing elections is not as firm as it is often made out to be, nor is the issue as rigidly partisan as often presented by political wonks. Kavanaugh should get his hearing and and face scrutiny, and moderate and progressive elements of both the Republican and Democratic parties should vote in opposition of the appointment if Kavanaugh does not demonstrate a commitment to upholding legitimate prior decisions made by the court, especially with regard to providing cover to a sitting president in the argument that the holder of the office should be shielded from prosecution, which is arguable an even more pressing and long term danger than overturning Roe v. Wade.