It's a "poem", everyone knows it, everyone is currently wallowing in it, it's bad. Why the scare quotes around "poem"? Although it is referred to in English as "First they came", it was not written to be or really intended to be a poem. It's a speech that he gave many times and was gradually transformed by him and others into what it now is. The history of how people wanted to read it has everything to do with its final form.
I should write at the outset that although Niemöller was an early supporter of Hitler, and an antisemite, I firmly believe in a separation of artist from art. It's possible for a bad person to make good art or a good person to make bad art. Criticism of who he was is not why it's bad art. There are so many reasons why it's bad that I'll have to put them into sections.
Aesthetic reasons
It's bad because it uses simple repetition for a sentimental effect.
Repetition is not in and of itself bad in poetry. There are plenty of good poems that use it. How is it used in this one? First they came for the X / And I did not speak out / Because I was not an X. For all of the variations of the English version of the poem (I'll get to the translation later) that is all the poem is for all of the verses but the last. You can see why people like it, it's very "accessible".
When a strong pattern is established by repetition, it's the repetition-breaking moment that establishes the effect. What's the last verse, which breaks the pattern? Then they came for me! No one left -- to speak out for me! The reader identification is strong, who has trouble identifying with a threat to -- me? (i.e. themselves).
I can't really bear to do a close reading of this, there is so little there. It wasn't intended as a poem and it barely functions as one. I will note that in what is referred to as the original German, the repetition is broken -- the first verse says "Nazis", and the others don't. This was taken out by whoever translated it for good aesthetic reasons -- to clean it up a little, making the repetition stronger, making the idea more general.
Textual variation
There's a well known part of criticism that goes through textual variations of works. For instance, Walt Whitman wrote different versions of various poems in _Leaves of Grass_, put them in different order etc. The textual variations abound here because it never was intended as a poem in the first place.
But we can learn something from them. First, the order in the initial speech was probably: Communists, disabled people, Jews, occupied countries. The first of these to get edited out were disabled people. Just not something people in a eugenics-influenced society reading it in English wanted to consider.
Then, of course, Communists out. Not very sympathetic especially for an American audience. Socialists and trade unionists in.
I should emphasize that I'm not sneaking biographical criticism back in this section, I have no idea how much of this was done by Niemöller modifying the poem himself to fit his audience and how much was other people. But it's actually changing the text of the poem, so it's important to the poem.
This is the worst kind of textual modification for this poem. Who will speak out for -- wait, that group got deleted.
Political reasons
Why is the poem bad politically? It is a political poem so this is a fair question to pose.
It's because it is written with an implicit protagonist who is an important person and who expects to be heard. The implicit protagonist in the poem (really Niemöller, but critics try to separate the person-in-the-poem from the author) evidently thinks that *speaking out* will do something. They came for the X -- but I did not speak out. What if the protagonist had spoken out?
Well you can try it now, in the current context of Trump. Trump is bad, he should not deport people, he should not try to eliminate disabled people or trans people, he should not suggest ethnic cleansing (in no particular order). Do not do that!
Did anything happen?
For the Iraq War, people didn't just speak -- they turned out in millions and peacefully protested.
Did anything happen?
What really causes political change, and what constitutes political resistance? The poem implicitly suggests a model that only powerful people can do, and that is probably ineffective for them even if they did it. That is a bad political model. It's a particularly bad model for anything like left or even centrist politics, which has to acknowledge that most people are not powerful and need to do more than speak out.
Moral reasons
Why is the poem morally bad? Again, it's a poem about morality so this seems like a fair area for criticism.
It's morally bad because it encourages reader identification that lets the reader see themselves as a basically good person who suffers from indifference and needs a reminder not to be indifferent *for their own good*. The morality in the poem is very pragmatic: who will be left to defend you?
In reality, the reader is probably not a good person. Societies don't go fascist with the majority of people remaining pure and good. Moral tracts that encourage people to see themselves as good are not themselves good.
This is really why Niemöller-as-Nazi keeps being discovered and undiscovered. It's not that he concealed his history from the people he started telling this to. They knew who he was. If anything, the poem is *stronger* in certain ways if you read it as from an early Nazi supporter.
But people don''t want to read the poem that way because that calls their own current goodness into question. They want to read it as: they are a good person, they have a choice of whether or not to speak out for people coming up, if they speak out they are good.
But they aren't if it's a Nazi sympathizer describing how he was a Nazi sympathizer. That puts the reader in as as a bad person and the speaking out part maybe only the beginning of clawing your way back.
Historical usage and reading
Why is the historical usage or reading of this poem bad? I'm going to get some overlap with sections above here, but I think it's worth looking at this as a separate question.
Poems accumulate preferred interpretations as part of the history of the poem. Robert Frost for instance wrote a poem for his friend who over-agonized about his personal decisions telling him those decisions didn't matter that much, and his friend took it the wrong way and decided to prove his bravery and got himself killed in a war.
This is a Cold War poem. It's all about the time when Germany had to consider its history, confront it and cry some tears, be welcomed as a bulwark of the West, and in general we had to congratulate ourselves as part of a bloc where we could speak out and that this freedom and the other freedoms guarded by a free people meant we should be on guard but basically this was never going to happen for us. So the historical usage or preferred reading was not to prepare against the return of fascism, it was more subtly one of reassurance.
How do you read the poem as a Jew, or as a Communist (oops -- socialist), or as a disabled person (wait they were deleted)? In that case I guess you are supposed to be reassured that people are being told to speak out for you in an ineffective way in their own self interest. But the poem is not for you.
Biographical criticism
I've concluded that I have to do some biographical criticism after all, because it's crucial to understanding the poem. Niemöller was not exactly a Nazi. He was a WW I U-boat captain who became a religious leader. He supported the Nazis early but differed from them exactly when they started to attack Jews who had converted to Lutheranism and joined his church. He was put in a concentration camp for 8 years as a VIP prisoner and later parlayed his fame into world-wide tours as a pacifist and human rights campaigner. He was someone who could declare "Hitler betrayed me" and mean that Hitler had a personal conversation with him in which Hitler made a promise to him that he later broke.
The mystique of this constellation of biographical events is part of the enduring popularity of the poem. The theme of "first they come for the despised, but they never stop there" is reinforced and proved by this leader-y quality.
I don't like, as a person and a critic, this idea of the exaltation of the ex doer of bad deeds who repents. It's a Christian cliche, and personally I find it disagreeable.
The current Palestinian genocide is going to produce, decades later, many Israelis who speak movingly about the horrors they inflicted as if they didn't know at the time or were carried away. I preemptively wish that they would shut up and not try to become celebrities: they knew, and they were not carried away.
Conclusion
It's an aesthetically, politically, and morally bad poem that has been altered textually in bad ways, read in bad ways, and has been badly interpreted.
It's not good.