pseudo-Anosov maps on surfaces with boundary In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed orientable surface, $f \colon S \to S$ an automorphism and $c$ is a geodesic 1-submanifold of $S$ such that $f(c) \simeq c$ then $f$ is reducible map.
Suppose $S = T^2 \sharp D^2 \sharp D^2$ (the twice punctured torus) and $\delta$ is a loop around one of the boundary components. Then $\delta$ is non-trivial in $H_1(S, \mathbb{Z})$ but $\forall [\phi] \in \mathcal{MCG}(S)$, $\phi(\delta) \simeq \delta$ or $\phi(\phi(\delta)) \simeq \delta$. Hence this statement doesn't hold for $S$.

Is there a similar result for $S$ (or indeed general surfaces with 2 or more boundary components)?

 A: First off, the mapping class group of a surface with boundary is generally taken to mean the group of diffeomorphisms that fix the boundary, up to isotopies fixing the boundary.  In this context, a Dehn twist around one boundary component is not isotopic to the identity.  What you're describing is usually called the mapping class group of a surface with punctures.
Under many definitions (including Casson and Bleiler's) the identity automorphism is also reducible (as well as periodic).  So I don't see a counterexample in what you write.
If you're asking, for which simple closed curves is the Dehn twist around that curve not the identity (and also not periodic), the answer is simple: it's those curves that do not surround a single puncture.  These are the same as the curves that are homotopic to a geodesic in any hyperbolic representative with cusps at the punctures.  (For most curves, you look at a shortest representative in its homotopy class and you automatically get a geodesic.  For a curve that surrounds a puncture, there is no shortest representative: you can make the curve arbitrarily short by pushing it out the cusp.)
