Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$.

The trick to get the Noether current consists in making the variation local: the standard argument, which does't convince me and for which I'd like a more formal explanation, is that, since the global symmetry is in force, the *only* term appearing in the variation will be proportional to *derivatives* of $\epsilon,$ and thus the involved current $J^\mu$ will be conserved on shell:
$$ \delta S = \int \mathrm{d}^n x \ J^\mu \partial_\mu \epsilon $$

This is stated, e.g., in *Superstring Theory: Volume 1* by Green Schwarz Witten on page 69 and *The Quantum Theory of Fields, Volume 1* by Weinberg on page 307.

In other words, why a term $ \int \mathrm{d}^n x \ K(x) \epsilon(x)$ is forbidden?

----------

Taking from the answer below, I believe two nice references are

 1. [theorem 4.1][1]
 2. [example 2.2.5][2]


  [1]: http://arxiv.org/abs/hep-th/0002245
  [2]: http://arxiv.org/abs/math/0108160