The cleanest answer, given the absurdity of the entire situation (a starship like Enterprise tumbling through the atmosphere without incinerating or coming apart?), is that Enterprise's shape was intermittently acting as an air brake as it tumbled through the atmosphere. As such, the people inside it would be "falling" toward a slower-moving Enterprise from time to time as she tumbled, slowed, then continued rotating and sped back up.

However, most of the scenes of people falling were inaccurate, as Enterprise would fall most slowly (and so the people inside would fall fastest) when she is falling near-parallel to her own Z-axis (that is, right-side-up or upside-down), and would fall faster (with the people inside moving more slowly relative to her) in attitudes where hallways were vertical and thus long, "dangerous" pitfalls. Most of the panicked people falling along hallways wouldn't be falling all that fast relative to Enterprise, and the impact would most likely have been with the floor or ceiling a few feet away. The big thing to worry about is falling away from a hallway or catwalk in a tall space, like Engineering or the main concourse-type area featured in a few scenes, and then falling down the "right" way as Enterprise tumbled through a level attitude.

But, out-of-universe in their defense, it's really hard to get a bunch of actors to fall the same way other than by natural gravity. Back in the TV shows, the bridge cast spent hours practicing and shooting fake sideways lunges choreographed with an opposing camera shake, and that was pretty much the limit of this type of thing in any ST show or movie up until the Abrams-verse (not counting the weightless scenes aboard Kronos I in ST VI, which were primarily wire work with CG blood). The shots from the hotel hallway scenes of Inception, which are stunning examples of how to play with gravity in-camera. required the set to be built in a rolling motorized cylinder. Not sure how they did the weightless scenes of the hotel set; there's a history of those kind of shots being done in a "vomit comet" but it could as easily have been wire work and digital enhancement. While these tricks wouldn't have been impossible for some of the Star Trek scenes, you also have to add in blue-screen elements which in turn would require a lot of complex (expensive) tracking of the virtual scene behind, so with a lather large CG budget to begin with, except for a few "hero shots" they faked it as much as they could.

The cleanest answer, given the absurdity of the entire situation (a starship like Enterprise tumbling through the atmosphere without incinerating or coming apart?), is that Enterprise's saucer shape was intermittently acting as an air brake as she tumbled through the atmosphere. As such, the people inside would be "falling" toward a slower-moving Enterprise from time to time as she tumbled right-side up, slowed, then continued rotating to the side or front and sped back up. When Enterprise is more or less right-side-up, that's also when she's falling slowest (presenting the largest cross-section to her direction of movement), and so you'd actually get some sensation of gravity, though most likely not full gravity.

However, given this explanation, most of the scenes of people falling were inaccurate, as Enterprise would fall most slowly (and so the people inside would fall fastest) when she is falling near-parallel to her own Z-axis (that is, right-side-up or upside-down), and would fall faster (with the people inside moving more slowly relative to her) in attitudes where hallways were vertical and thus long, "dangerous" pitfalls. Most of the panicked people falling along hallways wouldn't be falling all that fast relative to Enterprise, and any impact would either be with a wall at relatively low speed, or with the floor or ceiling just a few feet away. The big thing to worry about is falling out away from a hallway or catwalk in a tall space, like Engineering or the main concourse-type area featured in a few scenes, and then falling down the "right" way as Enterprise tumbled through a level attitude.

But, out-of-universe in their defense, it takes a lot of shot planning and a lot more film fakery to make this type of thing look good. Consider Inception, specifically the hotel hallway scenes (which are stunning examples of playing with gravity in frame). For the van-tumbling scene where the hotel's pull of gravity keeps changing, they put the set on motorized rollers and turned it through 720 degrees with the camera matching the rotation. I don't know how they did the fully weightless scenes as the van dives into the water, but it's not inconceivable that they put a few of the smaller set elements on a "vomit comet" and the rest were wire rigs and digital trickery. None of this is out of the realm of possibility for any of the Star Trek shots, but most of them were of much bigger spaces, like the main concourse and cargo bay, and others, while more enclosed, were longer, and nearly every hallway in Enterprise is curved for the obvious reason. My guess is that they did quite a bit of blue-screen roller-hallway work, and larger set elements on hydraulics, but there's just no way you're going to exactly simulate the environment inside a starship tumbling through the atmosphere except, well, by being in a starship tumbling through the atmosphere. You're certainly not going to be able to get your actors to do any appreciable acting while they're busy keeping their feet.