As written, Kronecker's statement [the one quoted by Gerry] is very similar to the idea of doing analysis in "weak" formal systems, such as first order Peano arithmetic, primitive recursive arithmetic, hereditarily finite ZF, or based on informal principles that roughly correspond to such systems. That is, "completed infinite" constructs (such as the totality of Cauchy sequences, or a single unspecified Cauchy sequence, or manipulations of infinite sets) may have a nebulous status but are OK as a heuristic or a formalism as long as in each case, such as formulas involving $e$ and $\pi$, the resulting manipulations are seen to ultimately reduce to calculations that can be performed and proved in PA, or whatever the acceptable theory of finitary constructions.

This is interpreting the posted question, "what *might* have been the intended meaning of Kronecker", as *what is a sensible reading of Kronecker's statement*, and not as the historical question of what he truly, originally, demonstrably did intend.

(Added: to put this another way, Kronecker might say that $\pi$ is fine as a formalism for organizing finite computations about explicit finitely presented objects such as algebraic numbers or matrices with integer entries, but that $\pi$ by itself is more ontologically dubious. Transcendence proofs, from this perspective, are a sequence of estimates about finite integer calculations, similar to irrationality estimates of $\sqrt{d}$ using continued fractions, which are interpretable as Diophantine inequalities between positive integers. There is some sense to this insofar as, even with a modern theory of algorithmic objects --- finite computer programs such as those computing $\pi$ approximations --- $\pi$ is inevitably a higher-type object than integers; verifying any given integer formula such as 2+2=4 is a finite calculation but $e^{i \pi} = -1$ requires some form of induction, no matter how explicitly one constructs $e$ and $\pi$.)