The classical version of the Lyapunov direct method is difficult to be applied for stochastic differential equations containing delay in the state variable. Criteria are presented which are relatively easy to verify when the
th moment exponential stability of the solutions is used. The results of Liu and Xia regarding the exponential stability in mean square are generalized to the
th moment stability,
. The paper contains a lemma proving some estimates of the solutions when the
th norm is used. The main results are presented in two theorems extending the estimates already obtained and proving sufficient conditions for
th moment exponential stability. All proofs are explicitly given and a patient reader will understand the subject. Some numerical examples are added.