Acceleration does not kill us any more than speed. If your head
and feet do not move at the same velocity long enough, whatever the
cause, you are in trouble. Velocity does not kill us when the whole body
has the same velocity.
Similarly, I doubt acceleration kills us when all parts of the body
accelerate, but without having to transmit forces. It is said in a
comment:
It's not the fall that kills you; it's the sudden stop at the end.
The sudden stop kills you because the deceleration (negative
acceleration) that stops you is actually caused by a force transmitted
through your body which cannot withstand it. The acceleration
throughout the fall, no matter how strong, which applies uniformly to
the whole body will not hurt it: you are in free fall.
If the same acceleration were produced by the pull of an engine
attached to your feets and pulling your whole body (even without
friction), rather than gravity applied uniformly to every atom of your
body, your body could well be torn to pieces.
I am no expert on jerk, but I somehow doubt that it is any more
danger, despite contrary statements in this accepted answer and this comment
The human body uses bones and muscles to maintain its integrity while
transmitting forces. The problem of jerk is that it changes the values
of forces, thus requiring muscles to adapt constantly.
But free fall satellite motion does have jerk, since the direction
of gravity is constanly changing, and its magnitude depends on
distance. This is generally true of non uniform gravity field.
I think, a good way of understanding what can hurt us is to model the
human body as two masses, head and feet, joined with a spring. If the
distance between the masses changes by more than, say, 5%, the human
model is considered dead. Now, if you add a strong structure, some
kind of G-suit, that forcibly preserve the distance between head and
feet, thus carrying all forces that need to be transmitted, then the
human model is pretty safe.
Note that submitting the head and feet to
different acceleration can have undesirable effects if the difference is important. But if the body
is strong enough, it can sustain small differences which it compensate with internal cohesion forces.
So one might say that speed can be more dangerous than acceleration,
when it is an issue of uniformity across the body.
To place these issues on the level of personal experience: we do not feel
speed, but we do not feel acceleration either, or jerk. What we do experience
is forces propagating through our body, when our body accelerate
because it is submitted to forces applied only to some parts of it,
rather than uniformly. We experience the tension of the muscles that
preserve our body structure against these forces. And we perceive jerk
as a need to adapt muscle tension.
When we say you are experiencing an acceleration this means something must be exerting a force on you, because force and acceleration are related by Newton's second law. In a stationary elevator it is the floor of the accelerator that exerts an upwards force on you, and this force is just $mg$ giving you an acceleration of $g$.
If the elevator is accelerating downwards, for example $-0.5g$, the force exerted on you by the floor of the elevator is decreased and your total acceleration is decreased (in this case to $+0.5g$). If the downward acceleration of the elevator is $-1g$ then the force on you decreases to zero and you become weightless i.e. your acceleration is zero $g$.
If the acceleration of the elevator becomes greater (more negative) than $-1g$ you will find yourself standing on the roof of the elevator so it is now the roof that exerts a net downwards force on you. This is how your acceleration can become negative. Instead of the elevator floor accelerating you upwards the elevator roof accelerates you downwards.
Best Answer
If the elevator is not accelerating, there is no way for the person in the elevator to know whether it's moving or still. So the laws of physics will be exactly the same: they jump with a certain force, reach a certain height, and land again.
If the elevator is accelerating (upwards), then the person will feel heavy: they will have difficulty jumping as high as usual (just as it was easy for astronauts on the moon to jump really high, as gravity was less there). So for the same "extra" force, they will appear to rise less high in the elevator. This is of course because, from an external observer's point of view, the elevator is accelerating up towards the passenger. But for the passenger it just feels like somebody "boosted gravity".
The reverse happens if the elevator is accelerating down (in the limiting case, the elevator could be in free fall). Now the passenger feels lighter, and can jump higher than usual. Until the free fall case, where the passenger will float (briefly, depending on the height of the building).