In a political sense - the focus here - a politician's base is one's core support, the people who are with you whatever may come ... or at least to a further point than most people.
It is not an entirely new term, even in this context, and the larger sense of it has some generally obvious meanings. A military base is the place where troops are stationed, armaments and other secure supplies and materials are located and to which - in a combat area - forces can retreat or collect in comparative security. In a loose sense, its a localized home territory, a refuge.
Some of that carries over into politics: One's political base is a political refuge of sorts.
The term got its biggest push in that context in 2004, when the George W. Bush campaign, seeking re-election, calculated it was more likely to get the votes needed to win (in the right places) by pushing hardest for turnout within the core support - the base - rather than by trying to reach across to pull in broader support.
Bush's campaign strategist, Matthew Dowd, recalled in an interview figuring how many persuadable independents were out there as opposed to how many already-sympathetic non-voters: "nobody had ever approached an election that I've looked at over the last 50 years, where base motivation was important as swing, which is how we approached it. We didn't say, "Base motivation is what we're going to do, and that's all we're doing." We said, "Both are important, but we shouldn't be putting 80 percent of our resources into persuasion and 20 percent into base motivation," which is basically what had been happening up until that point -- look at this graph, look at the history, look what's happened in this country. And obviously that decision influenced everything that we did."
That decision was significant in the management of presidential campaigns; it was the first explicitly to focus as much or toward toward maximizing the hard-core support as opposed to broadening support. That has had significant effects on both sides of fence, chiefly by deepening the political canyon between them. Many people in whatever remains of the middle are left stranded, and many on either side see the opposition in ever darker terms. During the Donald Trump presidency, notably, the effort please the base has specifically meant displeasing or even damaging everyone else.
Are we far from the point where being considered a part of a "base" becomes less a badge of honor with a small group than a criticism - a perjorative - with the large remaining majority? Maybe not far. And why not?