It is said here that "In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set theory was becoming the centre of controversy. Kronecker, who was on the editorial staff of Crelle's Journal, was unhappy about the revolutionary new ideas contained in Cantor's paper". However the authors of the entry don't elaborate on reasons why Kronecker was so unhappy, which prompted my curiosity.
About Kronecker's views on "foundational" issues, you can see:
- Anne Troelstra & Dirk van Dalen, Constructivism in mathematics: An Introduction. Volume 1 (1988), page 17.
There is no "explicit" philosophy of mathematics in Kronecker's works.
He may be regarded as a "constructivist", or perhaps as a precursor of the finitist approach.
In his essay "Uber den Zahlbegriff" (1887) he outlined the project of "arithmetizing" Algebra and Analysis; that is, to found these disciplines on the fundamental notion of number, avoiding thus geometrical intuition.
In his arithmetization project he considered a mathematical definition acceptable only if it could be checked in a finite number of steps, criticizing the "pure" existence proofs. He stated that an existence proof for a number could be considered correct only if it contained a method to find the number whose existence was proven.
Some of his remarks belong to mathematical folklore, like his widely reported statement that:
"the Lord made the natural numbers (ganze Zahlen), everything else is the work of men";
the same idea is reiterated in the following statement:
"I consider mathematics only as an abstraction of the arithmetical reality".
Note: for some references to Kronecker's opposition to Cantorian set theory, we can see José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Birkhäuser (2007), sub voce.
In conclusion, it seems that Kronecker never published his critique of Cantor's theory, but we can conjecture from the sources above that he disliked Cantor's new theory of the "arithmetics of the infinite" as well as Cantor's wide use of "pure" existence proofs.