It's not a bad idea in the abstract, but it's also not a trivial and cheap project. The author's example of Japan leaves out the little fact that although it's used on some new lines, and is being retrofitted on some old ones, the vast majority of Japanese rail/subway lines do not use platform barriers, and it will certainly be many decades, if ever, before a significant number do.
Besides the obvious construction costs, to make platform barriers work you need very consistent door positioning, which is maybe not so hard for new systems, but is a much bigger problem for old systems like the NYC subway (with multiple types of running stock, etc).
NYC has the additional problem that it has much higher (up to an order of magnitude in some cases) construction costs for public works projects than most other countries.
On the positive, platform doors are not an "all or nothing" proposition ... they can be added line-by-line, or station-by-station, as soon as necessary conditions are in place.
Certainly it's not a bad idea to study for the future, but it's also certainly not a simple project to undertake, and even if started soon, it would be a long time before enough any significant portion of the system is protected.
The 7 train is still controlled entirely manually, with the old block-based signalling mechanism that can only be aware of the location of a train in very vague terms. It's currently having CBTC (Communications-Based Train Control) installed, which will give dispatchers much higher resolution information about the location of every train, and go a very long way towards automating the entire system. CBTC installation is scheduled to be completed in 2016, and it's the reason why there are currently so many service disruptions afflicting it during weekends (it actually just began a series of 13 consecutive weekends during which it will not go into Manhattan). If I recall correctly from the posters, the total budget of the project is about $750 million.
If and when the line is fully automated someday, it will become possible to create a reliable barrier mechanism (like what currently exists on the very small automated portions of the NYC subway, which is really just the JFK AirTrain). Until then, human operators can't reliably get the train cars to stop in the same place.
Note that it certainly is possible to consistently do very accurate stops with only unassisted human operators; Japanese rail operators have done this as a matter of course, AFAIK using simple visual alignment aids, for as long as I can remember. [Traditionally it's done because people are lined up at the door locations (marked on the platforms), not for platform barriers.]
It's more a matter of operating culture and training than technology.
[Of course changing the latter may be easier in NYC!]
This doesn't sound right. I live in NYC, and people getting killed by trains is generally a newsworthy event. It's not something that one hears about every couple of days, as would be the case if it happened to 200 people every year. Also, for the fraction of these deaths that are suicides (which probably far exceed the number of people being deliberately pushed), building barriers on the platforms will not save these people's lives, since there are many alternate ways of committing suicide.
The NYC subway has 468 stations, with the larger ones (e.g., Times Square) having a dozen or more platforms. Add to that the commuter rail lines (e.g., LIRR, PATH, Metro North) and you have even more (Penn Station and Grand Central are huge). It would probably cost billions of dollars to build gates on every platform. These billions of dollars could probably save many more than 200 lives a year if, for example, they were spent on improving the health of poor children in NYC. (But in any case, the city doesn't have billions of extra dollars lying around.)
Yes, you would get badly banged up (watch the 3rd rail!) but would probably stand a better chance. I'd also like to verify the average actual cause of death before fully backing this idea.