The error is in the route, I should create parameter route like:

Route::get('dropdowns/stations/{id}', 'DropDownController@getStations');

And modify a bit the jquery script and DropDownController:

public function getStations($id)
	{
		$stations = Station::where('basin_id', '=', $id)->get();
		$options = array();

		foreach ($stations as $station) {
			$options += array($station->id => $station->name);
		}

		return Response::json($options);
	}

AJAX call:

<script type="text/javascript">
	$(document).ready(function() {
		$("#basin_id").change(function() {
			$.getJSON("../dropdowns/stations/" + $("#basin_id").val(), function(data) {
				var $stations = $("#station_id");
				$stations.empty();
				$.each(data, function(index, value) {
					$stations.append('<option value="' + index +'">' + value + '</option>');
				});
			$("#station_id").trigger("change"); /* trigger next drop down list not in the example */
			});
		});
	});
</script>