Back when I was in graduate school in the 1990s, the standard reference for this sort of thing was Kolb and Turner's book The Early Universe. Even after all these years, that book's treatment of this subject is probably still a good place to look.
Even if there's no asymmetry-producing process for neutrinos (like baryogenesis), you still expect a relic neutrino background that's a thermal (Fermi-Dirac) distribution of both neutrinos and antineutrinos, with a temperature of about 2 K. The reason is that, at a certain time in the evolution of the Universe, the density dropped low enough that the neutrino number "froze out": interactions that could change the number of neutrinos (such as primarily $e^- e^+ \leftrightarrow \nu_e\ \bar\nu_e$) became so rare that the time for any given particle to undergo such a reaction grew much longer than a Hubble time.
It's been a long time since I looked at baryogenesis models with any care, but as I recall some models would be expected to produce an asymmetry in the neutrino sector as well. But in practice I don't think that would change the prediction much. The reason is that baryogenesis only has to produce a one part in $10^9$ asymmetry (a billion and one protons for every billion antiprotons). That produces very noticeable effects today, because there was essentially complete annihilation of the antiprotons. But neutrino freeze-out occurs much earlier, while neutrinos are still relativistic, so we don't think that that massive annihilation happened for neutrinos. So even if there is a neutrino-antineutrino asymmetry comparable to the asymmetry produced by baryogenesis, it should only result in a tiny difference in the number of neutrinos over antineutrinos.
Let me put that another way. At early times, (temperature much greater than the proton mass), there were comparable numbers of photons, neutrinos, and protons. Baryogenesis resulted in an asymmetry of protons over antiprotons at that time. After that, nearly all of the protons and antiprotons annihilated, leaving the observed result that today there are a billion photons for every proton. But we expect the number of relic neutrinos to be of the same order as the number of photons, not protons, so a baryogenesis-level neutrino asymmetry won't be noticeable.