About Kronecker's views on "foundational" issues, you can see:
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.