You can use this event:

It is enough to add the placeholder attribute to this event:

$(this).data('select2').$dropdown.find(':input.select2-search__field').attr('placeholder', 'My Placeholder')

$('select').select2({
    placeholder: 'Select an option'
}).on('select2:opening', function(e) {
    $(this).data('select2').$dropdown.find(':input.select2-search__field').attr('placeholder', 'My Placeholder')
})

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<link href="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.3/css/select2.min.css" rel="stylesheet" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.3/js/select2.min.js"></script>


<select style="width: 100%;">
    <option value="volvo">Volvo</option>
    <option value="saab">Saab</option>
    <option value="mercedes">Mercedes</option>
    <option value="audi">Audi</option>
</select>

I had the same need and ended up writing a small extension for the Select2 plugin.

The plugin has a new option searchInputPlaceholder for setting a placeholder for the Search input field.

Add the following code after the plugin’s js file:

(function($) {

    var Defaults = $.fn.select2.amd.require('select2/defaults');

    $.extend(Defaults.defaults, {
        searchInputPlaceholder: ''
    });

    var SearchDropdown = $.fn.select2.amd.require('select2/dropdown/search');

    var _renderSearchDropdown = SearchDropdown.prototype.render;

    SearchDropdown.prototype.render = function(decorated) {

        // invoke parent method
        var $rendered = _renderSearchDropdown.apply(this, Array.prototype.slice.apply(arguments));

        this.$search.attr('placeholder', this.options.get('searchInputPlaceholder'));

        return $rendered;
    };

})(window.jQuery);

usage:

Initialize the select2 plugin using the searchInputPlaceholder option:

$("select").select2({
    // options 
    searchInputPlaceholder: 'My custom placeholder...'
});

Demo:

Demo can be found at JsFiddle.

Updated May 9, 2020

Tested with the latest Select2 version (v4.0.13) - JsFiddle.

Github Repository:

https://github.com/andreivictor/select2-searchInputPlaceholder