The politicization of the Supreme Court, though reprehensible, has been a foregone conclusion for a several years. Although Democrats taking the long view say a filibuster of Judge Gorsuch is not in Democrats' own interest, their base demands it.
Writing at The New Republic, Graham Vyse argues that Democrat Senators will pay no political penalty for doing what they can to obstruct the path of Judge Gorsuch to the Supreme Court. If "no penalty" means no one who voted for them before is likely to vote against them next time because of it, I agree.
On the other hand, will a filibuster garner for them any votes they didn't get before? Or win votes for Democrat challengers of sitting Republican Senators? I doubt both very much.
The sad state of today's Democratic Party has been widely documented and needs no recap here. They need to figure out how to win back the statehouses and House seats they've lost in the last 8 years. The filibuster isn't going to make Democrats more attractive to people who switched to Trump after staying home or voting for Obama in prior cycles.
The bottom line for Democrats: A filibuster won't make things worse. However, it also won't help in areas where they need to improve their image. Using the classic hole-digging analogy, it won't dig the hole deeper but it also won't help them climb out - their current need.