No. The scale of human or any animal activity on earth is small relative to the entire planet ecosystem. When you see the Earth from space, it's almost impossible to see any creatures living there or their effect on the planet.
In a big city where you are, you might not believe this, but if you could be taken on a trip to the nearby wilderness jungles and go deep within, or if you could go north to the wastelands of the Himalayas and further north to Mongolia or Siberia, or south to the outback of Australia, or for that matter on a sailing trip across the Pacific, you would see that the vast majority of the planet is empty and open and desolate. There's plenty of air and everything else.
Indeed, because of human activity, subtle changes are occurring in the atmosphere, in particular due to burning oil and gas, but the Earth is a huge dynamic system that compensates. Usually the changes are slow and take hundreds or thousands of years, so individuals, and even individual societies will not notice them in their lifetime. It's not something you should worry about. (Although the hole in the ozone layer was caused by humans, and that happened over several decades--so there are some rare disastrous things that can happen.)
To quantify things, take the burning of oil. Billions of barrels are being burned at an increasing rate. This has added to the amount of carbon dioxide in the atmosphere, raising it from about 300 parts per million to about 400 parts per million. While in relative terms, this is a huge increase, in the grand scheme of things, the absolute amount of carbon dioxide is still a minuscule fraction of the total atmosphere, much much less than 1%, so plenty of air or oxygen remains to breathe. Even if you could burn every single drop of oil on the planet there would be more than enough oxygen. As for deforestation, this is happening, true, but the vast majority of planetary oxygen comes not from forests, but from green plants in the ocean, mostly algae. Tremendous global changes would have to occur to present a threat to human or animal life, and if this does happen it is more likely to be from an asteroid hitting the Earth than from any human activity. This does happen about every 100 million years, and you might want to hope that it does not happen while you are alive.
Fortunately, loss of the oxygen in the atmosphere that sustains us is not a concern. Oxygen comprises 21% of the atmosphere, and most of us could live quite well with a few percent less. People's respiration is not significant as a contribution to the atmosphere's oxygen concentration, even with population increase.
Assuming present and anticipated rates of burning fossil fuels, estimates are that, even if no oxygen was produced by photosynthesis, it would take thousands of years for the oxygen concentration to fall, say two percent. We would have plenty of time to react.
Interestingly, if oxygen levels were to rise even one percent we would likely see a substantial increase in forest and grass fires.