The answer on such problem is following.
No anti- in the case of Majorana particle
First, if we include the mass term (here it is not important - Majorana or Dirac) of neutrino, then we necessarily come to the statement that the neutrino can be of two helicities.
Second, since Majorana fermion is $$ \nu_{M} = \nu + \nu^{c}, $$ where $c$ operation is charge conjugation, then fermion and antifermion are the same. Thus we don't distinguish neutrino and antineutrino in the case if neutrino is Majorana fermion.
There is also one additional remark. The charge conjugation of left particle gives right antiparticle, so there is no problem with chirality.
Why Majorana nature of neutrino doesn't contradict the experiment
We thus follow to the statement that neutrino, if it is Majorana one, may be as well left-handed as right-handed. This, however, doesn't contradict the statement that mostly only left-handed (in a sense left-helicity) neutrino interacts (and thus somehow may be detected). This is true because of two facts.
First, if the mass of given particle $\nu$ is small, then this particle is relativistic in most processes which involve it and other particles with masses much larger than $m_{\nu}$. In relativistic limit helicity coincides with chirality; namely, helicity flipping rate is proportional to $\left(\frac{m_{\nu}}{E_{\nu}}\right)^{2}$, with $E_{\nu}$ being the energy of $\nu$. We may therefore conclude, that ultrarelativistic particle has definite helicity which is defined by its chirality.
Second, the Standard model weak sector contains interaction, in which electron interacts only with left-chiral neutrino (while the positron interacts with right chiral). Let's unify these facts. If we have process in which electron and neutrino creates, then due to small neutrino mass and thus its ultrarelativistic motion it will be left-handed (in a sense of helicity) in "99.9%" of interactions.