Specifically phosphorylates seryl-tRNA(Sec) to O-phosphoseryl-tRNA(Sec), an activated intermediate for selenocysteine biosynthesis. ATP + L-seryl-tRNA(Sec) = ADP + O-phospho-L-seryl-tRNA(Sec) Aminoacyl-tRNA biosynthesis; selenocysteinyl-tRNA(Sec) biosynthesis; selenocysteinyl-tRNA(Sec) from L-seryl-tRNA(Sec) (archaeal/eukaryal route): step 1/2. Belongs to the L-seryl-tRNA(Sec) kinase family.