This is the timeline of The Terminator (1984). In this timeline:
- Skynet becomes self-aware in 1997 and "Judgement Day" follows, in which most of humanity is wiped out
- A war between the human survivors and the machines ensues
- Sarah Connor's son John is the commander of the human resistance forces; he sends soldier Kyle Reese from 2029 to 1984 to save Sarah from a T-800 that the machines had sent to kill her (so that John is never born)
- A predestination loop is formed, because Kyle becomes John's father
- Kyle saves Sarah from the T-800 (but dies in the process)
If we only consider the films and exclude all other material, then nothing is known about Sarah Connor's demise in this timeline. We only know that she gives birth to John, Skynet's self-awareness and "Judgement Day" still happen in 1997, and John goes on to command the resistance forces. (Whether she is alive in 2029 is irrelevant to the plot because the machines are trying to kill her before John is born.)
However, The New John Connor Chronicles continues the T1 timeline, and it is revealed that:
In the original unaltered timeline where Judgment Day occurred in 1997, Sarah was killed fighting Terminators in Buenos Aires in 2012.
A similar fate for Sarah in the original timeline is mentioned in the Randall Frakes novelization of Terminator 2, but Mexico is the location instead of Argentina.
Again, it depends on what one considers canon in the Terminator franchise.
This timeline is that of Terminator 2 (1991). This timeline starts off the same as Timeline #1, except that additional Terminators are sent back in time to 1995, after the events of The Terminator. A T-1000 is sent by the machines to kill John Connor when he is 10 years old. A reprogrammed T-800 is sent by the future John Connor to ensure the survival of his younger self. The T-800 successfully protects John, and John, his mother, and the T-800 destroy the Skynet project in its infancy, at least postponing "Judgement Day" (if not averting it completely).
If you believe that Terminator 3: Rise of the Machines is still canon and part of this timeline, then Sarah Connor dies in 1997 from leukaemia.
As @Richard points out in comments above, this is most likely from exposure to the nuclear cores of the T-800s. From this point of view, she possibly died in the late 1990s from this exposure in both Timeline #1 and Timeline #2, regardless of the validity of Terminator 3. (Also, we do not see Sarah in Terminator: Salvation, but again, we may be free to disregard that film now.)
There is also an alternate ending to T2, where we see that Judgement Day did not occur and Sarah is alive and well on August 29, 2029:
In the alternate ending, we also see that John Connor is a US Senator. On top of this, we have the Malibu Comics continuation of the T2 timeline (called the T2 Trilogy), in which the following happens:
After the destruction of the headquarter of Cyberdyne System, Sarah lives with the alias "Suzanne Kreiger" in Paraguay...Sarah meets Dieter von Rossbach at some point and eventually falls in love with him. She later marries him.
Sarah is still alive in 2029 and witnesses the departure of both Kyle Reese and the "Uncle Bob".
This is the timeline of Terminator: Genisys. This timeline starts off the same as Timeline #1. However, when Kyle Reese is sent back to 1984, he finds himself in a situation that is very different from when he was sent back in Timeline #1.
In this timeline, Sarah and Kyle Reese travel from 1984 to 2017 using a homemade version of the temporal displacement device constructed by a T-800 known as "Pops". There, they defeat John Connor (in the form of a human-Terminator hybrid designated the T-3000), and stop Skynet from proliferating itself as the Genisys operating system.
The Sarah Connor of this timeline has less exposure to damaged Terminators (for most of her life, "Pops" is completely intact, and any Terminators in pursuit of her are quickly disposed of). Also, if she were to develop leukaemia anyway, it would occur in the years following 2017, because of her time shift with Kyle Reese, meaning she might have access to better medicine.
So, in this timeline, we cannot say anything about her fate — we can only extrapolate from what we see in Genisys.