Another answer is based on the explicit representation of loop momentum amplitudes in string theory. It's for instance decribed in D'Hoker and Phong 1988 (the review) and is at the heart of the "chiral splitting" formalism. Long story short, when you're given a $g$-loop string integrand, you can undo a Gaussian integral so as to make appear explicitly the loop momentum. This will give you expression that possess a term like: $$\exp(-\sum_{I,J=1}^g \ell_I \Im \Omega_{IJ}\ell_j)$$ where $\ell_I$ are the $g$ loop momenta and $\Omega_{IJ}$ is the so-called period matrix, of size $g\times g$, which generalises the complex structure $\tau$ at one-loop (which is a one-by-one matrix in this case).

And now you may use modular invariance to see that $\tau$ is always non-zero and larger than some finite constant ($\Im \tau>\sqrt{3}/2$) (same sort of relations on $\Omega$), so that large $\ell$ produce an exponential suppression of the loop integrand.

Another answer is based on the explicit representation of loop momentum amplitudes in string theory. It's for instance described in D'Hoker and Phong 1988 (the review) and is at the heart of the "chiral splitting" formalism. Long story short, when you're given a $g$-loop string integrand, you can undo a Gaussian integral so as to make appear explicitly the loop momentum. This will give you expression that possess a term like: $$\exp(-\sum_{I,J=1}^g \ell_I \Im \Omega_{IJ}\ell_j)$$ where $\ell_I$ are the $g$ loop momenta and $\Omega_{IJ}$ is the so-called period matrix, of size $g\times g$, which generalises the complex structure $\tau$ at one-loop (which is a one-by-one matrix in this case).

And now you may use modular invariance to see that $\tau$ is always non-zero and larger than some finite constant ($\Im \tau>\sqrt{3}/2$) (same sort of relations on $\Omega$), so that large $\ell$ produce an exponential suppression of the loop integrand.