New York has low turnout because it has a set of laws that serve to disenfranchise voters in ways that other states could only ever dream of. (Yes, people technically have the right to vote, but because of a set of laws which I've described elsewhere, there's no way for them to use these votes to hold their elected officials accountable, like there is in other states).
Once every decade or so, yes, there's a race that's moderately contested (like the one you describe). But at that point, people aren't in the habit of voting anyway.
By the way, in your example the only reason Teachout ran as a Democrat instead of on the Working Families Party as she'd planned is because of these laws. The Working Families Party was pressured into endorsing Cuomo as their candidate, to ensure that they'd retain their ballot access. Teachout was angry enough that she decided to run as a Democrat, knowing full well that she'd lose, because nobody in New York ever wins a primary without the backing of the major party[0].
Teachout is far and away an exception, not the rule. Even then, her run was purely symbolic, and she herself knew it.
[0] Again, due to a whole bunch of laws that give the party incredible influence in shaping the outcome of primaries.