The actual answer is that it doesn't matter. For many of the 1:1 solid-state structures, either the cations or the anions may be considered to be at the vertices (i.e. corners) of the unit cell. By symmetry, both representations are entirely equivalent.
To see why this is the case, it is helpful to look at a 2D analogue first. The following could be considered to be a 2D version of the cesium chloride (CsCl) structure.* The blue and orange dots represent cations and anions respectively (or the other way round; it doesn't matter, the point is that they're different things).
From the way I've drawn the unit cells (i.e. the black lines joining the dots), it appears that the blue dots should be at the vertices and the orange dot in the centre of the unit cell. It's obvious that this is a valid unit cell, as you can repeat it as many times as you wish to generate the full crystal structure.
However, I could just as easily have drawn this:
Again, this is a valid unit cell; but this time, the orange dots are at the vertices and the blue dot is in the centre.
This shows that there is no difference between the two representations of the unit cell. Both of them can be repeated (and translated) to form exactly the same crystal structure. That should already suggest to you the answer to the 3D case in your question.
Let's look at the 3D CsCl structure. You can draw it out in 3D and try to convince yourself again using the argument above. However, I think it's easier to use the 2D depiction on the left, where the numbers indicate the positions of the atoms (lattice points) in the third dimension, as a fraction of the unit cell length in that dimension. So, 0 and 1 indicates that there is a blue atom at the front and at the back, whereas 0.5 indicates the orange atom right in the middle of the unit cell.
What we need to do is to extend this unit cell a couple of times in each dimension. The first arrow shows us how to extend the unit cell in the third dimension, which has been collapsed: essentially, we need to add 1 to every number (which corresponds to adding a new unit cell behind the current one). The second arrow is just a tiling in the other two dimensions.
I'm now going to redraw the unit cell:
Now, all we need to do is to subtract 0.5 from every number (this is OK because it's just shifting the zero, i.e. moving our "point of view" backwards / forwards) and we get the same CsCl unit cell but with the atoms swapped round.
A similar exercise may convince you of the NaCl / rock salt case.†
* That's similar to the body-centred cubic structure, but is not the same: in the bcc structure the atom in the middle is the same as the one at the edges. Thanks to Karsten Theis for pointing that out.
† Similarly, the NaCl structure is similar to the face-centred cubic, but fcc has the same atom at all the lattice points.