Select list elements based on other list

 Select[listA,MemberQ[listB,#[[1]]]&]  

{{"a", 1, 3}, {"e", 4, 4}}

You can also use Cases:

Cases[{Alternatives @@ listB, __}] @ listA

{{"a", 1, 3}, {"e", 4, 4}}

You can use an association:

take = Lookup[AssociationThread[#[[All, 1]] -> #], #2, Nothing] &;
take[listA, listB]
(* {{"a", 1, 3}, {"e", 4, 4}} *)