This piece is a nice one on the perplexities that Democrats face on abortion.  This is not really structurally or functionally parallel to the problems faced by the GOP, and it's also important to see that the GOP and the Democrats are not identical with "the right" and "the left" in the US.  But it is perhaps the one issue that is most fraught among Democrats today.  

I say "fraught" because the issue threatens to divide supporters of the Democrats--not just divide Democrats from Independents--in ways that could cause trouble for the party.  Data suggests a fairly stable level of support for abortion rights, across many decades now, but that same data masks tensions within the support (and also tensions within the GOP about opposing such rights). There are racial divisions on this issue, notably among Hispanics in the Pew data. 

But the polling data only tells a small part of the story.  What we're seeing is this: over the past fifty years, there has been a sustained effort to cultivate a "pro-life" movement in the US, which means (imho) fundamentally an anti-abortion rights movement.  (Efforts to create a larger, what one might call more genuine, "pro-life" movement have been, well, abortive.).  This movement has not met with a parallel opposing cultural force.  The "pro-choice" movement has nowhere near the institutional infrastructure nor the direct communication capacities of the "pro-life" movement.  Interestingly, the data suggests the efforts have largely been a matter of treading water.  This isn't exactly failure; after all, many of the same people who are working to convince people to oppose abortion rights are also opposed to same-sex marriage and other forms of moral change that have seen enormous levels of social change over the past half century, so maybe their efforts have staunched the bleeding.  But no more than that.  

And now, the anti-abortion rights movement is increasingly tied to support for this president, who is as pollsters say, "historically unpopular."  In fact the hard core of Trump supporters look to be about the same amount of the US populace as the anti-abortion rights people--each about 40%.  Is some sort of connection, beyond coincidence here?  It's unclear.  But I suspect, without much beyond my intuitions (which could translate into english as "wishful thinking") that this will do a great deal of damage, especially among younger voters, to the "pro-life" cause.

What I predict will cause the Democrats (continued) heartburn, however, is that at the same time that anti-abortion rights supporters have seemed to become identified with this presidency, abortion rights activists have gained increasing agenda-setting power within the Democratic party, so that it seems harder for Democratic politicians to find any way to register disquiet with unconstrained abortion rights.  The Hyde amendment is problematic in terms of punishing poor women.  But it is hard to imagine someone today supporting (Bill) Clinton's argument that abortion should be "safe, legal, and rare."  Assuming that there is a reasonably sizeable constituency of potential Democratic voters who would support that, it might prove a challenge to the party in the coming election cycles not to be able to speak to those voters.

UPDATE: about 45 minutes after I wrote the above, I came across this piece on the Washington Post website, which has some interesting data on all this.   And this piece in the New York Times makes the point as well.