I always find it remarkable that people who live in a city that is perhaps the country’s best example of the value of density are so skeptical of the value of density. Yes, increasing the number of people who live in an area will generate some downsides. It will also generate benefits, to local residents and society as a whole. At some point the marginal downsides entirely offset the marginal benefits and it no longer makes sense to build in a given place, but there is no indication that New York is anywhere close to this level.
Megan complicates things:
[Adding significantly more people to NYC] requires not just changing zoning rules--as far as I know, there's already quite a lot of real estate in the outer boroughs that could accommodate more people, but it's not close to transportation, so it's not economically viable. If you want to add a lot more housing units, you also need to add considerable complimentary infrastructure, starting with upgrading the rest of the subway's Depression-era switching systems (complicated and VERY expensive because unlike other systems, New York's trains run 24/7). And ultimately, it's going to mean adding more subway lines, because short of building double-decker streets, there's no other way for enough people to move.