Most people don't know where the railroads are in the Adirondacks, much less in New York State or the rest of the United States. On the East Coast, the biggest name in passenger rail operators is Amtrak, who still operates two lines very near the Adirondacks.
Looking just at New York State there are two main lines: one North-South line and one East-West line which intersect at Albany, NY. The two lines go by a variety of names based on the trains that are running, but the North South line seems to be generally known as the "Adirondack" line while the East-West line is known as the "Empire Service" (Amtrak site). As the blue line runs to the edge of Lake Champlain, the Adirondack line does pass through the park.
However, Amtrak takes advantage of only a little of the available rail in New York State. There are many more lines, as shown by this map:
The inset map in the upper left show the intercity lines operated by Amtrak. Focusing on the Adirondacks, there are only a few railroads which actually go through the park. There is one line which goes North out of Saratoga Springs operated by the Saratoga and North Creek Railway and there is a main line operated by CSX transportation (a freight mover) along the Northern border of the park that also has serveral spurs run by other operators. The last line is the one that is the source of the current debate: the Remsen-Lake Placid Travel Corridor ("the Corridor"). From the look of the abandoned rail lines, this corridor might have once traversed the park.
It is worthwhile to make a note about who actually owns the rails (which is not always the operator). Looking at this map of the taxable status of railsroads, also put out by DOT, it is pretty clear that the corridor is in a class of its own along with most of the commuter rail operated around New York City as they are owned by the State. Amtrak does not own the main lines that it operates on, these are owned by different freight movers as shown in the map on page 36 of the State Rail Master Plan.
The final thing of importance in these maps is the Saratoga and North Creek Railway. From the look of their website, they operate a tourist train not much different from the one operated on the Corridor by the Adirondack Scenic Railroad. However, their line is not state owned (see map of taxable status of railroads linked to above) so they don't have the same issues with the laws surrounding state land. In many ways their service is similar to one that a rail line along the whole corridor might provide: access to one of the two Adirondack ski areas (Gore for the SNC and White Face for the corridor), and a business which seems, from the look of their website and schedule, to be working (website)