There used be a time when there were very few differences between Democrats and Republicans, and most of those differences were limited to social liberties. Both parties were in favor of supporting the middle class
This I think is the crux of the problem, and it's not limited to the US, it's a global phenomenon.
What happened was that the Berlin Wall fell, and with it communism. Then socialism, the more moderate and acceptable face of the political left, was tainted by association. The left moved ever rightwards in an attempt to remain electable during and in the aftermath of the 'greed is good' Reagan/Thatcher years. So the two main parties, once espousing ideologically very distinct philosophies, became more and more similar to one another. They both sought the same sort of voter. The left abandoned its traditional core support of the poor working class. After all, it didn't matter if they moved to the right: so long as they remained the left-most party, then the poor working class would still vote for them, right?
No, wrong. What happened was that the poor became disillusioned and disenfranchised. No-one represented their interests any more. They had no-one to vote for. And increasingly, as politics move rightwards 'the poor' came to encompass more and more people, including the young, the lower-middle-classes, etc.
A political void opened up, and into this stepped the populists like Trump, policy-light but always ready with media-friendly soundbites, claiming to represent the average voter, those 'left behind' by the political mainstream. And they did this by appealing to people's base prejudices. It's very simple but tremendously effective. Where the mainstream career politicians are hamstrung by the necessities of diplomacy, the populists face no such constraints. Indeed their freedom from constraints is what makes them appealing. Trump can blame everything on the Mexicans, he can build a wall to keep them out, he can demand that his Democrat opponent be locked up... and on and on. And he rode that populist wave all the way to the White House. Trump's administration is government by pithy soundbite; whether there is substance behind these utterances or not is immaterial, the soundbite itself is the message. If Obama or George W had come out with even one of the things that Trump says every five minutes, there would be outrage and calls for them to step down... but Trump doesn't have this, the outrage is part of what empowers him: because it is the outrage of the establishment, and he had positioned himself as an outsider. It's a huge strength; nothing they say can hurt him. This is why the impeachment was a stupid idea; he was guaranteed to be acquitted by a partisan Senate, and it then becomes very easy for him to frame the attempt to remove him as a last desperate gamble by the establishment. It just cements his power-base and even draws more voters towards him, the valiant underdog swimming through a sea of corruption.
To belatedly get back to the Democrats then, and the left generally across many western democracies, the political centre has become poisoned, and the outsiders are now seen by many as a best hope for a representative who is truly
representative, who stands up for the interests of the common man (and indeed woman). This I think is why Bernie's popularity is growing. At the same time it explains why the DNC are trying everything in their power to stop him winning: because the party and the voters that the party should represent have moved apart. The party is more centrist than the people who vote for them. Both parties are. The electorate is becoming more polarised, and the mainstream political class from both parties are becoming more similar to one another and, on occasion, indistinguishable from one another.
So what will happen?
I think on the one hand that politics is cyclic. If the DNC defeat Bernie, or Bernie wins but Trump beats him, then it's not the end of everything. Support for Trump will fade as ordinary people see that their living standards haven't risen. It took a while for them to lose faith in the mainstream politicians, and it will take them a while to lose faith in Trump, too. But they will eventually, and centrism will likely come around again as the popular option. Democrats will regroup and will win again.
On the other hand, if Bernie wins and then gets the presidency, and is able somehow to get some of his policies enacted, then this is a huge opportunity for the US to shift the whole global narrative leftwards, away from the 1980s Reagan legacy and towards a future where inequality is reined in and government genuinely cares for and represents the welfare and the best interests of the electorate.
Either way, yes, I am very confident that the Democrats will survive. But what we mean when we say 'Democrats' may shift a little bit from what it is now. Politics is nothing if not malleable.