In the Astronomy Picture of the Day from August 8, 2020 ("Crescent Saturn"), the caption states
From Earth, Saturn never shows a crescent phase
Why is this? We can observe phases on planets such as Venus which have an apparent size a lot smaller than Saturn. Saturn is not tidally locked to the sun (only Mercury is).
So why does the sunlit face of Saturn always face Earth?
Phases are just different perceived illuminations of an object at different illumination and observing angles. If the observer is, with respect to the object, located in a similar direction as the light source shining on the object then you should expect to see the vast majority of the object illuminated, if the observer is located in the opposite direction you would see the object back-lit, and if you are at right angles observing the object with respect to the direction of the light source you would see the object half-illuminated.
Since Mercury and Venus are always inside Earth's orbit and move with different rates around the Sun as Earth does, the Earth (the observer) is able to locate itself at any angle with respect to the light source that shines on the observed planet. This means that you can see any phase of Mercury or Venus (except from a perfect 100% illuminated phase due to the body of the Sun blocking the view). Here you have an example for Venus:
Now think about what happens from the point of view of an Earth's observer for the exterior planets. The Earth will never have a chance to see the planet's back-lit side since there's no position in its orbit that would allow for this. As seen from the other planet, the Earth is always close to the Sun, so it can be seen almost exclusively during the day, which means that from the Earth you almost exclusively see illuminated regions of the planet.
Mars is the closest exterior planet, so the Earth manages to gain enough elongation to see a bit of the night side, but it is a tiny fraction of the disk as viewed from here. Here you have a picture of the phases of Mars as seen from the Earth:
This is even worse for far away planets. Saturn is so far away that from its vantage point, that the Earth is basically always close to the Sun (to the light source). From the Earth, Saturn seems always fully illuminated, by an extremely small margin that allows to see a slim crescent of darkness in perfect conditions. Only with spacecraft like Cassini and Voyager have we been able to see what Saturn looks like from behind. The first time humanity did this was in 1980 with photographs like this one (from Voyager):
Before that we had never seen the night side covering more than a percent of the disk. Even the shadow of Saturn cast over the rings is nearly impossible to spot from the Earth. Look at this amateur photograph that shows precisely a bit of that shadow over the rings behind Saturn (bottom-right part):
That shadow is almost non-existent from Earth's point of view, and it's all because Earth's orbit is inside Saturn's and Saturn is far away from the Earth. The Earth is always so close to the line connecting the light source (the Sun) and the illuminated object (Saturn), that you shouldn't expect any more phases than "full" from here. Simple geometry.
All the other answers here are complete, and more in-depth than anything I would write. However, if you prefer to look at things visually, here is a terrible not-to-scale 2 minute paint drawing.
No matter where earth is in it's orbit, you'll never be able to see more than a sliver of the dark side of Saturn as we're too (relatively) close to the sun.
The answer is simple geometry: we are 1 AU from the Sun, Saturn is 10x further from the Sun. Looked at Earth from Saturn, Earth is always in front, behind or immediately next to the sun; an observer at Saturn would see phases of Earth, similarly as Mercury appears to us.
Argued the other way around, we are so close to the Sun that we always only see the day side of Saturn as we can never be at a similar distance so that we never see the terminator or the night side (as Cassini could image it).
As a side note: tidally locked is unrelated to which side is illuminated (see our Moon) - unless a body is tidally locked to the Sun. We do see different sides of Saturn as it rotates - it just is not as apparent as with Jupiter as the cloud formations on Saturn are not as distinctive as on Jupiter.
The crescent phase is only visible from a location further away from the Sun than the object. See for example this diagram of lunar phases: the crescent phases occur when the Moon is on the sunward (relative to the Earth) side of its orbit.
The Earth is always closer to the Sun than Saturn is. Therefore you can never see Saturn in a crescent phase from the Earth.
I made a little animation to demonstrate what others have already said. Here the outer planet is at 3 AU (Saturn is around 9 AU, much further). The light side of each planet always faces the sun. Since the earth is so close to the sun relative to Saturn, we can pretty much only see its light side. On the other hand, if you were looking at Earth from Saturn, you would see phases: when Earth is between Saturn and the sun, you see the dark side. On the right-hand side you can see the what each planet would approximately look like from the other.
