Many answers already make good points. I don't think there's only one answer to this -- it depends on the context and particularly the student. I think if a student seems to be able to make good progress initially without the labels, it's perhaps better to avoid them. But...
A few points that haven't been highlighted so far:
- Students with no pre-existing knowledge of music notation will probably benefit more from having note names labeled somehow. The combination of trying to learn to decode basic music notation at the same time as learning the piano keyboard can be a bit overwhelming. For those who already can read music, there's less of an initial cognitive load -- just learning the placement of notes on the keyboard.
- Younger kids in particular may enjoy the "matching game" of being able to match a key to a note, particularly if an initial book has note names as part of the notation. This is more likely to lead beginners to experiment and try out pieces in a book, even if they are not part of an official "lesson." A lot of the difficulty with young children is keeping them interested and not frustrated, so anything that allows them to learn things, play around, or try out stuff by themselves is potentially a good thing.
- The frustration problem is also an issue with older learners, particularly those are trying to self-teach. New piano students are already trying to learn how to make their hands move, how to strike the keys properly, etc. And on top of that, they are trying to decode music notation and learn the location of keys. For some people, having note labels more readily available can alleviate at least some of that frustration initially. Particularly for the self-taught, creating a new habit and learning an instrument by oneself is already a big commitment. Finding ways to make it easier to get started can be helpful. Without a teacher there to correct you, it can be harder to verify you're hitting the right notes in the first place.
Also, I'm not sure I agree with those who deemphasize letter names for notes (or who would substitute colors or shapes, except perhaps for very young children who don't know letters yet fluently). The theory seems to be that one should make a direct connection between the visual location of the note on the page and the physical location on the keyboard; the note name is secondary.
But I think that can be its own sort of "crutch." We have a name for music notation that shows one how to physically arrange one's fingers on an instrument -- it's called tablature. There used to be tablature for keyboard instruments, as well as various other instruments, though nowadays it's mostly seen for guitar. The standard staff music notation we use is meant to be a somewhat universal system (at least for Western music), which can be helpful for any instrument and music in general. Part of that is associating the abstract names (i.e., letters) with particular notes.
I've seen way too many sophomore (even junior) music majors in college who still needed to write down note letter names next to notes to be able to read music properly, even though they've played an instrument or sung for over a decade. (I've even seen said students have to draw a picture of a keyboard on a page to decode what notes were which, before they could identify a chord or something.) Avoiding letter names can create its own problems in the long run, assuming the student ever wants to talk about music with anyone else, learn any music theory, or understand music written for anything other than piano. If one wants to use shapes or colors, I think that's fine if combined with letter names in the shape or color so students will eventually make the letter-name connections. If you're going to use a crutch, why not make it a useful and educational crutch?
I also agree with all of those who note that labeled can also be a crutch that can take a while to wean a student off of. But sometimes one needs a crutch to get over the initial difficulty. (Think of how many young piano students quit after a year or three, many because they simply have difficulty decoding music notation and matching it to the keyboard.) There's no single answer for which is better in the long-run -- I think the needs and personality of the individual student should factor in.