It's difficult to answer accurately without seeing your markup, but maybe you can use a multiple selector to alleviate code redundancy. You can also use prop() to enable or disable your elements more easily:

$(function() {
    var $elements = $("#amount, #period, #key");
    $elements.prop("disabled", true);
    $("#api").change(function() {
        $elements.prop("disabled", !$("#apiEnabled").is(":checked"));
    });
});

try fallowing way-

$(function(){    
$('#amount').attr('disabled','disabled');
            $('#period , #key').attr('disabled','disabled');
            $('#api').change(function(){
                if($('#apiEnabled').is(':checked')){
                    $('#amount, #period , #key').attr('disabled','');
                }else{
                    $('#amount, #period , #key').attr('disabled','disabled');
                }
            });
});

This is my take at it, but if you won't be needing the separate elements all over the place, then take @Frédéric Hamidi's advice and create a single object.

$(function(){
    var $amount     = $('#amount'),     // cache jQuery objects
        $period     = $('#period'),     // to avoid overhad
        $key        = $('#key'),        // of fetching them
        $apiEnabled = $('#apiEnabled'); // all the time
    $amount.prop('disabled', true);
    $period.prop('disabled', true);
    $key.attr('disabled', true);
    $('#api').change(function(){
        if ($apiEnabled.is(':checked')) {
            $amount.prop('disabled', false);
            $period.prop('disabled', false);
            $key.prop('disabled', false);
        } else{
            $amount.prop('disabled', true);
            $period.prop('disabled', true);
            $key.attr('disabled', true);
        }
    });
});

You can cache your set of elements to prevent jQuery from looking them up every time you refer to them. You can also group them all into one set because you are applying the same code to all three of them.

prop can be used instead of attr which is new in jQuery 1.6+. prop is better suited to checkboxes and setting things as disabled.

Now that all three elements are grouped, you have one place (the checkboxes variable) that you can add or remove new checkboxes. It also means more succinct code in your change event.

$(function(){

    // Cache the elements
    var checkboxes = $("#amount, #period, #key");

    // Disable them using prop rather than attr (jquery 1.6+)
    checkboxes.prop('disabled', true);

    $("#api").change(function(){

        if($('#apiEnabled').is(':checked')) checkboxes.prop('disabled', false);
        else checkboxes.prop('disabled', true);

    });

});

$(function(){    
    var $els = $('#amount, #period, #key');
    $('#api').change(function(){
        var unchecked = !this.checked;
        $els.each(function(){
            $(this).prop('disabled', unchecked);
        });
    });
});

And if applicable change:

var $els = $('#amount, #period, #key');

To:

var $els = $('#api').siblings();

Example Here:

http://jsfiddle.net/BpVkw/