How much mass per year may the black hole at the center of the Milky Way gain by capturing neutrinos? If we are immersed in a sea of neutrino, then a black hole should be constantly capturing the ones crossing its event horizon, and gaining mass.

How much neutrino mass falls inside a black hole per year? For example, the one at the center of our galaxy.
Not enough to worry about.
Most neutrinos we detect on Earth come from the sun.  The black hole at the centre of the galaxy is not so close to a star as the Earth is so you would expect the neutrino flux to be lower than on Earth,  let's assume it is similar: about $10^{11}$ neutrinos per $\text{cm}^2$ per second. But each neutrino is light, even including its kinetic energy it has only a few hundred KeV (again based off solar neutrinos for the sake of a rough estimate). Converting that to mass ($m= E/c^2$) gives the mass of neutrinos per cm² per second.  It's about $5×10^{-20}$ kg/cm²/s
The black hole is relatively small in diameter, even if you account for the fact that the effecive radius is larger than the actual event horizon radius, since things will tend to be bent towards the black hole, the cross sectional area is about 10²⁵ cm², so roughly 500000 kg of neutrinos might fall into the black hole every second.  (Although this is very much in the spirit of a Fermi estimate, this is probably an overestimate)
But the mass of the black hole is about 8,000,000,000,000,000,000,000,000,000,000,000,000kg
So 500,000kg is negligible.