As someone who lived in New York City through most of the Eighties, I had a good look at Hizzoner, who died yesterday at the ripe age of 89 or so. He was a very typical Homo Politico of his era (not referring to his sexuality; I'll get to that later); a hard worker, totally committed to his personal political ambition, full of bad motives but not financially corrupt, neither all good nor all evil. He was a man of longstanding grudges, most of them against fellow Democrats.
In terms of his lasting legacy, I'd say look at NYC, and specifically Manhattan, of today--if you like it, he showed the way for it to change from the unruly, dangerous place it was in the Seventies to the gentrified, safe (for wealthy whites), stopped-and-frisked (if you're a minority) theme park of the 21st century. He started the transformation with his 12-years in City Hall; then, after the brief, unhappy misdirection of David Dinkins' mayoralty, it returned to his path--law-and-order, fiscal sanity through challenge to the public unions--followed by Rudy Giuliani and Michael Bloomberg.
He was an incredibly popular mayor in New York, winning against divided opposition by huge margins, until he rather suddenly was not; then he got "primaried" and dumped. That was probably his bitterest moment, even worse than his loss in the 1982 New York gubernatorial primary to his arch-enemy, Mario Cuomo; that loss pretty much ended any ambition Koch had of going beyond the city limits, but the loss of control in the city in 1989 was a deeper blow. He responded by opposing Dinkins at every turn, helping to ensure his failure, and then backed Republican Rudy Giuliani in his successful challenge to D.D. four years later.
His endorsement of Giuliani started a pattern of bizarre and ill-considered endorsements. His guiding principle seemed to be that of personal enmity. He fell out with Giuliani, endorsed the odious Republican ratfink Senator Alfonse D'Amato, and, worst of all, opted for George W. Bush in 2004 (that, probably, was out of a vain hope to get a job in his second-term Administration; needless to say, it didn't happen).
His endorsement, like that of someone like David Duke or, nowadays, Donald Trump, was a tainted thing. In 1988, his dislike of the two leading candidates for the Democratic presidential nomination, Michael Dukakis and Jesse Jackson, led him to endorse Al Gore before the New York primary. I was supporting Gore, too, largely because I saw Dukakis and Jackson as sure losers and considered Gore a perfectly reasonable Democratic alternative (anticipating the moderate road Bill Clinton would take to victory in 1992), but Koch's backing put the sure stamp of stink on Gore's candidacy. Sure enough, his support in NY dropped like a rock, Dukakis won big, putting him on a clear path to the nomination, and Gore soon dropped out.
Koch's career as a Congressman from the city was before my time there; he seemed to be a standard-variety "limousine liberal" (in the memorable words of GeorgeWallace) who had successfully challenged the Democratic party machine in rising through Greenwich Village neighborhood organization. The turning point in his career from just another liberal House backbencher to a historical figure of significance was his bitter 1977 mayoral primary vs. Mario Cuomo. I was not there, but I have heard the tales of the cars roaming the streets with voices in bullhorns urging "Vote for Cuomo, not the Homo" (Cuomo, of course, disavowed the tactic). Koch narrowly won the primary, and his career hit the big time.
Koch was a lifelong bachelor who kept his private life private (bravo!), but it led to the rumors and their unkind, unsuccessful alleged exploitation by overzealous Cuomo backers. I would say the best description of him was asexual, turned on mostly by politics. He certainly didn't do ny favors to the gay community in the early days of the AIDS epidemic, when he was the mayor in New York and could have done more.
In sum, with his ups and downs, personal vendettas, notable accomplishments and failures, and his general shifty politics, I would describe him as a Democratic version of Richard Nixon. The difference was that Koch outlived his political career by a quarter century of near-irrelevance, while Nixon faded from the scene and died much more quickly.