One way to solve this, while still using your mazeSolver.answer approach, is to pass an extra parameter which tracks whether we're in a recursive call or not. If not, we reset the answer:

function mazeSolver(maze, index = [0, 0], top_level_call = true) {
  if (top_level_call) {
    mazeSolver.answer = "";
  }
  const pointer = () => maze[index[0]][index[1]];
  if (index[1] < maze[0].length - 1) {
    index[1] += 1;
    if (pointer() === " " || pointer() === "e") {
      mazeSolver.answer += "R";
      if (pointer() === "e") return mazeSolver.answer;
      return mazeSolver(maze, index, false);
    } else {
      if (index[0] < maze.length - 1) {
        index[1] -= 1;
        index[0] += 1;
        if (pointer() === " " || pointer() === "e") {
          mazeSolver.answer += "D";
          if (pointer() === "e") return mazeSolver.answer;
          return mazeSolver(maze, index, false);
        }
      }
    }
  } else if (index[0] < maze.length - 1) {
    index[0] += 1;
    if (pointer() === " " || pointer() === "e") {
      mazeSolver.answer += "D";
      if (pointer() === "e") return mazeSolver.answer;
      return mazeSolver(maze, index, false);
    }
  }
}
const mySmallMaze = [
  [" ", "*", " "],
  [" ", "*", " "],
  [" ", " ", "e"],
];
console.log(mazeSolver(mySmallMaze));

That said, a more common and standard approach would be to just pass around the answer itself as a method parameter:

function mazeSolver(maze, index = [0, 0], answer = "") {
    ...
    answer += "R";
    ...
    return mazeSolver(maze, index, answer);
    ...
    return answer

}

There are at least a couple reasons to prefer the second approach:

1. In the second approach, all the inputs needed by the recursive function are in one place: parameters. This is easier to understand than the first approach, where some inputs are in the parameters and one is stored as a property on the object itself.
2. Consider the behavior if we assign mazeSolver to another variable: let solver = mazeSolver. Now solver is storing data on mazeSolver, which is messy and counterintuitive.