I don't really believe that.
Yes there are problems that are bigger than they might be if we had less people, but there's no reason to assume that a lower population will inherently be better for the environment. For example, the United States represents 15% of all CO2 emissions (according to some sources) despite only having 4.3% of the global population. China is responsible for 28% of CO2 emissions and has 18.2% of the global population. And yet in many developed countries there's more forest and other similar environments than there were a century ago. Of course that doesn't make up for CO2 emissions. But the problem is not inherently one of population.
And even if the population was stable at a much smaller size, if the economy continued to grow and the energy use of our civilization continued to grow then we would see environmental damage over time, and eventually the damage would rival our current damage on a global scale (but on a per capita basis it would be much more damaging).
Lowering the population size is not a solution, I'm afraid. For one, many economies require a growing or at least stable workforce - this can be dealt with but that would take resources away from addressing world problems.
I would argue that it is easier and more effective to address world problems with an expanding population than with a shrinking population. An expanding population gives the species a larger and larger pool of labor from which to draw upon. Of course it can expand too fast as well.
I would also argue that meeting human needs are of higher importance than environmental issues. This is debatable, of course, and the environment is a major factor of human needs so preserving it is important even for that goal, so we should still do it.
It turns out that humans don't live on that much of the Earth, and as the global population urbanizes more and more I suspect that most of the inhabited land will become more scarcely inhabited. So what's happening to all the land? Well, suburban sprawl is an issue that needs to be addressed, though I don't have a solution. And of course there are projects and a lot of industrial processes and other human activities on uninhabited land - so the damage caused by that needs to be minimized. I haven't read a thorough analysis so I don't even know what's affecting what, but we can find out and address those issues. Agriculture is a major issue - though advanced technology could change this tremendously. One major industry that affects the environment in major ways is the meat industry - however the damage could be reduced tremendously if cultured meat could be developed and delivered to consumers. This can also help with the rising demand for meat and has ethical benefits - no more slaughtering required to get meat. This would also free up vast amounts of land and could potentially reduce demand for Amazon Rainforest clearing, as over 60% of the cleared land is used for pastures, which would become virtually unnecessary if we adopted cultured meat. This would also free up the crops that normally go to feed animals, allowing for more food to go to human beings. More efficient agriculture could be developed - such as indoor multi-level agriculture, which could reduce the need for vast amounts of agricultural land currently used for raising crops. With isolation of the agricultural areas the need for damaging insecticide and pesticide can be reduced tremendously and highly efficient aeroponics could be employed along with optimization of the environment for the plants to massively improve yields. If fish meat can be cultured as well then the demand for fishing could be reduced, and the overfishing problem could be mitigated.
Energy is a major aspect of carbon dioxide emissions. Currently renewables don't seem able to keep up with demand. Generally solar has a capacity factor of nearly 25% while wind reaches 40% averages. Meanwhile coal regularly goes above 60% and nuclear's average tends to stay at or above 90%. Not only this but the shorter lifecycles of renewable energy technology leads to other issues, such as a higher replacement rate. Add on to this the low capacity factor and much more capacity has to be installed than is demanded in order to meet demand. This means that renewables have hidden costs, not to mention some fairly nasty environmental effects - solar panel waste is known to be toxic (though this can be minimized) and since they have a higher replacement rate there would be quite a lot of waste that we would need to deal with. Space based solar power could work, though in the interim I recommend nuclear power. The main disadvantage of nuclear is its immense cost, which is well known. However much of this cost can be eliminated with proper project management and a more mature nuclear power industry. With advancements in heat engine technology (mainly by using supercritical CO2) efficiencies can be increased by fairly large amounts and costs potentially reduced (when compared to steam turbines). Renewables should be used but they need better energy storage solutions - one I've come across is referred to as PowerLoop, an idea by Keith Lofstrom (of Lofstrom Loop fame). This concept involves linked iron rotors undergoing electromagnetic acceleration through a track as a form of energy storage. Another advantage is that it can be fairly efficient at delivering energy over long distances. One major advantage over flywheels is that a PowerLoop system is not limited in the same way by the strength of the material. Another advantage is that as the technology matures it could be adapted for space launch as a LaunchLoop - which could provide benefits as well. Renewables are not quite ready (yet), though they can be and I would say should be used in conjunction with energy storage. However some areas can not use renewables, or not to the necessary scale. Where I live it is simply impractical. Nuclear has to be a part of our energy production (and it is for now). As such we need to have a varied energy grid.
Transportation is another major issue. Electric vehicles show potential but they have problems as well such as the toxic materials in the batteries among other concerns. However South Korea has developed a way to charge electric vehicles while they drive on a road. This would require digging up and replacing all current paved roads to employ, but it is a potential solution. Hydrogen can be employed in aircraft but it has practical issues. Of course these can be overcome but it will be difficult to replace existing infrastructure. One fairly major aspect of shipping's environmental effect is its emissions. Some alternatives to the current propulsion systems include nuclear (though this could have issues, but may be more mature than nuclear power thanks to the US Navy), and potentially hydrogen-electric or hydrogen internal combustion. Another issue is that personal transportation is arguably too prevalent in some countries. A much wider and more effective/efficient public transportation system would be immensely useful, especially if it is all electric.
And if that isn't enough then climate engineering may be necessary. This all almost completely theoretical, but it may be possible to engineer Earth to be better for us and the other lifeforms of this planet.
Science and technology can give us solutions to most, if not all, of the problems we face while enabling humanity to prosper as a species and save the biosphere.
Of course this is more short term in the grand scale. I sometimes wonder if our waste heat will ever grow big enough to rival the energy the Earth gets from Sun... If that happens I wonder if dynamic orbital rings could be used to suspend radiators in space around the planet to radiate the excess heat away. Now that'd be crazy. Imagine Earth with rings like Saturn - but these would have a practical purpose.