Suppose we have an Anosov diffeo $f$ on $M$. Define $g_t : M\times\mathbb R\to M\times\mathbb R$ by $(x,s)\mapsto (x,s+t)$. Take a quotient of $M\times\mathbb R$ under the relation $(x,s)\sim (f(x),s-1)$. The flow descends to a flow $f_t$ on the quotient manifold.
My question is : As the flow is locally a translation, how can it contract the stable direction? I see that a contraction does take place at regular intervals but is that enough?
The short answer, as pointed out in the comments, is that yes, contraction at regular intervals is enough, because the definition of Anosov requires that there are $C>0$ and $\lambda\in (0,1)$ such that $\|Df^n|_{E^s}\| \leq C \lambda^n$ for all $n\geq 0$, and similarly for $\|Df^{-n}|_{E^u}\|$. The constant $C$ allows the possibility that there are some points at which $E^s$ does not contract and/or $E^u$ does not expand under one iterate (or even several iterates) of $f$. However, if you take $n$ large enough that $C\lambda^n < 1$, then $n$ iterates is always enough to see the hyperbolicity, no matter where you start.
There's a subtlety in your question that is worth pointing out, though. The definition of Anosov involves a Riemannian metric (although for a compact manifold the property of being Anosov is in fact independent of the choice of metric). If we write $M_1$ for the quotient manifold, then in order to ask "is the flow $f_t$ Anosov on $M_1$?" we need to first choose a Riemannian metric to use on $M_1$. Even though $M_1$ is a quotient of $M\times \mathbb{R}$, which carries a natural Riemannian metric, you can't simply push the metric down under the quotient map $\pi \colon M\times \mathbb{R}\to M_1$, because $f$ is not an isometry, so pushing the metric from $y\in \pi^{-1}(x) \subset M\times \mathbb{R}$ to $x\in M_1$ may give a different result depending on which point $y\in \pi^{-1}(x)$ you choose. And if you try to make a single global choice, say by identifying $M_1$ with $M\times [0,1)$ so that $\pi$ becomes a bijection (which I suspect is the picture in your head when you say that the flow is "locally a translation"), then you wind up with a discontinuity at the points where you glue $(x,1)$ to $(f(x),0)$.
Thus it's not immediately obvious what Riemannian metric to use. One way to proceed is to write $h_0 \colon TM\times TM \to \mathbb{R}$ for the original metric on $M$, put $h_1 = h_0 \circ Df$, connect $h_0$ to $h_1$ by a family of metrics $h_t$ that varies smoothly in $t$, and then extend periodically to $\{h_t \}_{t\in\mathbb{R}}$, so that $h_{t+n} = h_t \circ Df^n$ for all $n\in\mathbb{Z}$. Using $h_t$ as the metric on $M\times \{t\}$ and declaring $T_{(x,t)}(M\times\{t\})$ to be orthogonal to $T_{(x,t)}(\{x\}\times \mathbb{R})$ gives a metric on $M\times\mathbb{R}$ on which $(x,t) \mapsto (f(x),t-1)$ acts isometrically, so this metric does descend to a metric on $M_1$. Then you can check that the Anosov condition holds.
