Mapping string into integers

For the specific numbering in OP, you can also use LetterNumber:

LetterNumber[l]

{{2, 3, 4}, {5, 2}, {1, 2, 4, 5}}

The following gives you a "reversed" version of your association, with keys and values flipped:

l1 = <|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>;

lookup = First /@ PositionIndex@l1
(* <|"a" -> 1, "b" -> 2, "c" -> 3, "d" -> 4, "e" -> 5|> *)

You can then use ReplaceAll (/.) to do the replacement:

l = {{"b", "c", "d"}, {"e", "b"}, {"a", "b", "d", "e"}};

l /. lookup
(* {{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} *)

Alternatives

Other possible solutions for creating lookup:

lookup = <|Reverse /@ Normal@l1|>
lookup = Reverse /@ Normal@l1 (* doesn't need to be an association *)

l /. Reverse /@ Normal[l1]

(* {{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} *)

or

l /. AssociationMap[Reverse, l1]

(* {{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} *)