I'm trying to prove that the classifying space of the mapping class group of a surface $S$ with genus $g$ and $n$ boundary components is homotopy equivalent to the moduli space $\mathcal{M}_{g,n}$. The easiest proof I can think of is since $\mathcal{M}_{g,n}\cong \textrm{Teich}(g,n)/MCG(g,n)$ the moduli space is necessarily a $K(MCG(g,n),1)$ if Teich$(g,n)$ is a contractible CW-complex.
I certainly know it's true for negative euler characteristic, in which case Teich$(g,n)\cong \mathbb{R}^{6g-6+2n}$. Is it true for general $g$ and $n$ though?
Thanks for any help
First, according to the dimension of the Teichmueller space, it seems you are considering $n$ punctures and not $n$ boundary components. The real dimension of the Teichmueller space of a surface of genus $g$ with $b$ boundary components and $n$ punctures is $6g-6+3b+2n$.
Assuming that you are taking $b=0$ in your case, you need to worry about Euler characteristic zero case, i.e. $2-2g-n=0$, i.e. cases $g=1$, $n=0$, in which case the Teichmueller space is homeomorphic to $\mathbb{H}^2$, and $g=0$ and $n=0,1$, in which case the Teichmueller space is trivial. So they are contractible.
There is one issue, though: the mapping class group does not act freely on Teichmueller space. Stabilizers of points are automorphism groups of Riemann surfaces which are finite (in fact, there is a general upper bound given by $84(g-1)$, for closed surfaces), but may be non-trivial. For $(g,n)=(1,0),(2,0)$ the hyperelliptic involution fixes the isotopy class of every simple closed curve and thus fixes all points in Teichmueller space. Similarly, any surface obtained from a hyperelliptic surface of genus 2 by pinching simple curves to punctures, i.e. $(g,n)=(1,2),(1,1),(0,4),(0,3)$ will have a non-trivial stabilizer. For $g \geq 3$, generically the action is free, but there are examples where the automorphism group is non-trivial (for example, Klein's quartic achieves the maximum order of 168 for an automorphism group of surface genus 3).
So, since the action of $MCG$ has torsion, $\mathcal{M}_{g,n}$ can't be a $K(G,1)$ space even if $\operatorname{Teich}(g,n)$ is contractible. There is, however, a finite index non-torsion subgroup $\Gamma \subset MCG$ which allows to make the construction you want, and then you can consider cohomology with rational coefficients to relate the construction to $\mathcal{M}_{g,n}$. You can find the details of this construction in Chapter 12 of Primer on Mapping Class Groups, by B. Farb and D. Margalit.