In the beginning (or at least the beginning for me, I was pretty young during the Koch era) there was Dinkins, and the city was a crime ridden cesspool. You weren't supposed to take the subway at night, or go anywhere near a park. Parked cars routinely had their radios and sometimes tires stolen. There were very aggressive beggars all over the place. Whole parts of Manhattan, much less the outer boros, were no go zones.
Then came Giuliani. I think most people that lived in the city under Giuliani have very mixed feelings. The crime rate dropped, which was great, but there was very much a sense that a deal had been made with the devil. While to this day stop and frisk is a major issue for minority communities, back then there was a sense that the NYPD was killing and beating up people for no reason. Then there was the whiff of corruption (around Kerik and others), the messy personal life, fights with the unions, and the drama around the Senate race. I'll skip the post-9/11 apotheosis because it's not really relevant in the long run.
So that's when Bloomberg came in. He continued to preside over a drop in crime, but without as much visible divisiveness (though as mentioned above stop and frisk is still very much alive). Ditto with the public sector unions, they were still unhappy but there didn't seem to be the same sense of crises surrounding labor affairs. Meanwhile the finances of the city are very much improved, in spite of looming pension and health care problems the city is in very good financial shape as compared to any other large American city. No one thinks he's corrupt -- people think he's too rich to be corrupt. Then there's the nanny stuff, there's mixed opinions about that, but even smokers seem to be okay with having no smoking in bars. The biggest knock on him is that he only cares about Manhattan and the rich people that live there. That may be true from an emotional standpoint, but from a practical standpoint what he's done has benefited all layers of the city. The gentrification of Manhattan has generated a ton of revenue which is being plowed into services across the city. Honestly other than being better at feeling people's pain, I don't really see the argument.