  In this paper we define coeffective de Rham cohomology for basic forms on a
$K$--contact or Sasakian manifold $M$ and we discuss its relation with usually
basic cohomology of $M$. When $M$ is of finite type (for instance it is
compact) several inequalities relating some basic coeffective numbers to
classical basic Betti numbers of $M$ are obtained. In the case of Sasakian
manifolds, we define and study coeffective Dolbeault and Bott-Chern
cohomologies for basic forms. Also, in this case, we prove some Hodge
decomposition theorems for coeffective basic de Rham cohomology, relating this
cohomology with coeffective basic Dolbeault or Bott-Chern cohomology. The
notions are introduced in a similar manner with the case of symplectic and
K\"ahler manifolds.
