In this work, we look for possible new physics effects on the electromagnetic charge and anapole form factors, and , for a massless Dirac neutrino, when these quantities are calculated in the context of an effective electroweak Yang-Mills theory, which induces the most general -invariant Lorentz tensor structure of nonrenormalizable type for the vertex. It is found that in this context, besides the standard model contribution, the additional contribution to and ( and , respectively) are gauge independent and finite functions of after adopting a renormalization scheme. These form factors, and , get contribution at the one-loop level only from the proper neutrino electromagnetic vertex. Besides, the relation (, ) is still fulfilled and hence the relation (, ) is gotten, just as in the standard model (SM). Using the experimental constraint on the anomalous vertex, a value for the additional contribution to the charge radius of is obtained, which is 1 order of magnitude lower than the SM value.