I have a source claiming that this book wasn't banned, and that it was the most common of books confiscated from children when they arrived at concentration camps. Yet, the Nazis banned Kästner from publishing any other books in Germany. So why did Emil and the Detectives survive? Surely they would have assumed that all of his work would fall under the same category, having being written by a 'morally decadent' author?
The Nazis' ideological reasons for banning certain authors and books were threefold:
“volksfremd”: “alien to the (German) people”, this included all Jewish authors but also Remarque's All Quiet on the Western Front which they felt betrayed the glorious memory of Germany's WWI veterans.
“entartet”: “degenerate”, this included books with overtly sexual themes, such as were prevalent in Kästner's other work that wasn't targeted at children. His novel Fabian, for instance, featured lots of sex, including gay sex and prostitution. It painted a less than rosy picture of German society in the early thirties. He also had a sharp satirical pen and an eye for man's weaknesses and failings.
left-wing politics
As Kästner wasn't Jewish, and not actively involved in left-wing politics, only his objectionable works were burned. He was prohibited from publishing because he was not allowed to join the official association of German authors that the Nazis established. However, the immensely popular Emil was not blacklisted and actively sought out for destruction as it was unobjectionable. Goebbels even had such admiration for his literary talent that he employed Kästner under a pseudonym to write film scripts and several plays that he wrote under an alias were turned into films. When Hitler found out, he ordered Goebbels to cut it out.
Sources:
Ine Van linthout. Das Buch in der nationalsozialistischen Propagandapolitik. de Gruyter, 2011
Book Burnings: How the Nazis Ruined Erich Kästner's Career in Der Spiegel, April 18, 2013