The following works on your example:

sinList = {Sin[2 x], Sin[3 x], Sin[4 x]};
f = 1 + 2 Sin[2 x] + 3 Sin[3 x] + 4 Sin[4 x];
Prepend[Map[Coefficient[f, #] &, sinList], f /. Thread[sinList -> 0]]

I want to get the coefficients of the belw polynomial 1+2sin(2x)+3sin(3x)+4sin(4x) as {1,2,3,4}.

But the above is not a polynomial? If you want to get {1,2,3,4} you can try

Clear[x];
f = 1 + 2 Sin[2 x] + 3 Sin[3 x] + 4 Sin[4 x];
f = List @@ f;
f /. Sin[_] -> Sequence @@ {}
(*{1, 2, 3, 4}*)

If the input series is "0 + 0 Sin[2 x] + 0 Sin[3 x] + 4 Sin[4 x];" Only get the result {4}

There is no bug. Mathematica input itself simplifies 0 Sin[2 x] + 0 Sin[3 x] to zero before even the evaluator gets hold of the expression; So if you type

  f = 0 + 0 Sin[2 x] + 0 Sin[3 x] + 4 Sin[4 x]

and look at f you'll see it is 4 Sin[4 x]

There is nothing you can do about this really, unless you use Hold and friends when you actually type the expression itself.

      f = Hold[0 + 0 Sin[2 x] + 0 Sin[3 x] + 4 Sin[4 x]]

But once you release the Hold, we are back to square one. May be you can try using Strings. But I am not sure what you are trying to do there.