Our mamalian ancestors took over the world after the dinosaurs were wiped out. They needed the ecological freedom to do so, just like the dinosaurs profited from the mass extinction at the end of the Permian, 245 million years ago. Though, to be a bit more precise, the dinosaurs became dominant only at the end of the Triassic when they replaced many of the vanishing amphibians, reptiles and mammal-like reptiles that had existed earlier.
How the world would look like from the outside if the dinosaurs hadn’t been wiped out, is easy enough to say: pretty much the way it does today, because plate tectonics would have moved the continents around the same way - which also means that many of the fundamental climate factors would have worked similarly with or without the dinosaurs.
But the composition of life on our planet would be different, for sure. And life has an enormous impact on the state of affairs on a planet, even on things like the composition of the atmosphere and the chemistry of the oceans (which, in return, influence the climate).
So, how would life look today? No idea. You see, life, species interact; they develop as much in competition as in cooperation – and both factors stabilize and change the status quo, just like the fundamental forces of evolution: selection and mutation.
We cannot say how the dinosaurs would have further developed and if they could have stayed as dominant in a colder climate, for instance, that was to come. Maybe mammalians would have seized the chance and maybe more dinosaur species would have become more mammalian-like. Maybe one species would have learned to see the benefits of controlled fire.
Without massive extinction events, the composition of life surely wouldn’t have been altered as radically as it happened on more than one occassion but it would have adapted to the longterm changes that are inevitable on a planet with a hot core, plate tectonics, oceans and an atmosphere. And dependant on a life-giver that behaves reliably but not totally stable.
Life would be different, very different, but it still would be. And might even know that it is.