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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 usuallybasic cohomology of $M$. When $M$ is of finite type (for instance it iscompact) several inequalities relating some basic coeffective numbers toclassical basic Betti numbers of $M$ are obtained. In the case of Sasakianmanifolds, we define and study coeffective Dolbeault and Bott-Cherncohomologies for basic forms. Also, in this case, we prove some Hodgedecomposition theorems for coeffective basic de Rham cohomology, relating thiscohomology with coeffective basic Dolbeault or Bott-Chern cohomology. Thenotions are introduced in a similar manner with the case of symplectic andK\"ahler manifolds.

Coeffective basic cohomologies of $K$--contact and Sasakian manifolds