  The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants
of its characteristic foliation F (the orbit foliation of the Reeb flow). We
show some fundamental properties of this cohomology, which are useful for its
computation. In the first part of the article, we show that the basic Hodge
numbers, the dimensions of the basic Dolbeault cohomology groups, only depend
on the isomorphism class of the underlying CR structure. Equivalently, we show
that the basic Hodge numbers are invariant under deformations of type I. This
result reduces their computation to the quasi-regular case. In the second part,
we show a basic version of the Carrell-Lieberman theorem relating the basic
Dolbeault cohomology of (M,F) to that of (C,F), where C is the union of closed
leaves of F. As a special case, if F has only finitely many closed leaves, then
the (p,q)-th basic Hodge numbers are zero for p not equal to q. Combining the
two results, we show that if M admits a nowhere vanishing CR vector field with
finitely many closed orbits, then the (p,q)-th basic Hodge numbers are zero for
p not equal to q. As an application of these results, we compute then the basic
Hodge numbers for deformations of homogeneous Sasakian manifolds.
