Monte Carlo integration is more efficient in higher dimensions. To see this, let us assume that our integration volume is a hypercube with side $L$ and dimension $d$. This cube contains hence $N=(L/h)^d$ points and therefore the error in the result scales as $N^{-k/d}$ for the traditional methods.

The error in the Monte carlo integration is however independent of $d$ and scales as $$\mathrm{error}\sim 1/\sqrt{N}.$$ Always!

Comparing this error with that of the traditional methods, shows that Monte Carlo integration is more efficient than an algorithm with error in powers of $k$ when $$d>2k.$$