It can be shown the sum over all Feynman diagrams in quantum field theory leads to a unitary S-matrix. Where can I find a proof that the sum over string worldsheets with varying genuses also leads to a unitary S-matrix?
A way to prove it is to establish the equivalence of this Riemann-surfaces formula with the formulae you get from the light-cone-gauge calculation of the S-matrix that is derived from a Hermitian Hamiltonian and is, therefore, manifestly unitary. See e.g. Chapter 11 (if I remember well) of Green-Schwarz-Witten for a flavor.
Alternatively, you may try to prove the unitarity directly in the Euclidean signature Riemann surfaces. You will be led to the formulae for gluing and cutting of the Riemann surfaces, see the end chapters of Volume I of Polchinski's String Theory for a flavor.
You may also make the proof as similar to the analogous proof in QFT as possible - by using string field theory (either light-cone gauge, as in the first paragraph above, or the covariant one). The action's being real is the key to the unitarity in the latter case.