How can I create an enumeration variable by groups?

r1 = Join[{Append[First[data], "Count"]}, Join @@ Values @ 
  GroupBy[Rest @data, First, MapIndexed[Join, #]&]]

{{ID,Value,Count},{1,48,1},{1,45,2},{1,52,3},{1,43,4},{1,41,5},{2,50,1},{2,42,2},{2,51,3},{2,52,4},{bb,52,1},{bb,54,2},{dd,20,1},{dd,25,2},{dd,27,3},{cc,30,1}}

Update 1:

Prepend[Join @@ (MapIndexed[Join, #]&/@ SplitBy[Rest @ data, First]),
 Append[First[data], "Count"]]

{{ID,Value,Count},{1,48,1},{1,45,2},{1,52,3},{1,43,4},{1,41,5},{2,50,1},{2,42,2},{2,51,3},{2,52,4},{bb,52,1},{bb,54,2},{dd,20,1},{dd,25,2},{dd,27,3},{cc,30,1}}

Update 2:

addCounter = Module[{cnt}, cnt[_String] := "Count"; cnt[_] := 1; {##, cnt[#]++} & @@@ #]&;

addCounter @ data

{{"ID", "Value", "Count"}, {1, 48, 1}, {1, 45, 2}, {1, 52, 3}, {1, 43, 4}, {1, 41, 5}, {2, 50, 1}, {2, 42, 2}, {2, 51, 3}, {2, 52, 4}, {bb, 52, 1}, {bb, 54, 2}, {dd, 20, 1}, {dd, 25, 2}, {dd, 27, 3}, {cc, 30, 1}}

Here's one possibility using Split (which assumes that the IDs being counted always appear in runs)

splitcount = 
  Transpose[
    Flatten[{Transpose@#, 
      {Flatten[{
         "Count", Range /@ Length /@ Split[#[[2 ;;, 1]]]
      }]}
    }, 1]
  ] &;

To do some time trials on longer data lists (though not as long as the one you're looking at), first build some data:

SeedRandom[123]
idlist = Flatten[
   ConstantArray[#, RandomInteger[{1, 100}]] & /@ Range[10000]];
vallist = RandomInteger[{1, 60}, Length@idlist];
data = Join[{{"ID", "Value"}}, Transpose[{idlist, vallist}]];
Length@data

(* 507939 *)

Then

AbsoluteTiming[
 res1 = splitcount[data];
 ]
AbsoluteTiming[
 (* @kglr *)
 res2 = Join[{Append[First[data], "Count"]}, 
    Join @@ Values@GroupBy[Rest@data, First, MapIndexed[Join, #] &]];
 ]
AbsoluteTiming[
 (* @kglr *)
 res3 = Prepend[
    Join @@ (MapIndexed[Join, #] & /@ SplitBy[Rest@data, First]), 
    Append[First[data], "Count"]];
 ]
AbsoluteTiming[
 (* @kglr *)
 res4 = addCounter@data;
 ]
AbsoluteTiming[
 (* @JasonB. *)
 keys = AssociationThread[Union[data[[2 ;;, 1]]] -> 0]; 
 tally = Join[{"Count"}, keys[#] += 1 & /@ data[[2 ;;, 1]]]; 
 res5 = MapThread[Append, {data, tally}];
 ]

res1 == res2 == res3 == res4 == res5

(* {0.416703, Null}
   {0.739995, Null}
   {1.16564, Null}
   {1.81539, Null}
   {1.93773, Null}
   True  *)

I can't speak to the efficiency of this, you'll have to try it on your dataset,

keys = AssociationThread[Union[data[[2 ;;, 1]]] -> 0];
tally = Join[{"Count"}, keys[#] += 1 & /@ data[[2 ;;, 1]]];
data2 = MapThread[Append, {data, tally}]

(* {{"ID", "Value", "Count"}, {1, 48, 1}, {1, 45, 2}, {1, 52, 
  3}, {1, 43, 4}, {1, 41, 5}, {2, 50, 1}, {2, 42, 2}, {2, 51, 3}, {2, 
  52, 4}, {bb, 52, 1}, {bb, 54, 2}, {dd, 20, 1}, {dd, 25, 2}, {dd, 27,
   3}, {cc, 30, 1}} *)