Well, I've tried several permutations of this but to no avail. The entire element as seen rendered in IE Developer is below.
I have put the css on my customized Seattle.html page (can't edit the .master!) at the top of the page right before the closing head tag like so:
<style type="text/css">
#ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_sboxdiv {
display:none;
</style>
I also tried using the #searchInputBox but no love. I am publishing a major version of the masterpage after each edit and then refreshing the browser.
<div class="ms-srch-sb ms-srch-sb-border" id="ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_sboxdiv"><input title="Search this site" class="ms-textSmall ms-srch-sb-prompt ms-helperText" id="ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_sbox"
accesskey="S" onkeydown="EnsureScriptFunc('Search.ClientControls.js', 'Srch.U', function() {var ctl = $find('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr');ctl.activateDefaultQuerySuggestionBehavior();})" onkeypress="EnsureScriptFunc('Search.ClientControls.js',
'Srch.U', function() {if (Srch.U.isEnterKey(String.fromCharCode(event.keyCode))) {$find('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr').search($get('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_sbox').value);return Srch.U.cancelEvent(event);}})"
onfocus="EnsureScriptFunc('Search.ClientControls.js', 'Srch.U', function() {var ctl = $find('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr');ctl.hidePrompt();ctl.setBorder(true);})" onblur="EnsureScriptFunc('Search.ClientControls.js',
'Srch.U', function() {var ctl = $find('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr'); if (ctl){ ctl.showPrompt(); ctl.setBorder(false);}})" type="text" maxlength="2048" value="Search this site" autocomplete="off"
autocorrect="off"><a title="Search" class="ms-srch-sb-searchLink" id="ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_SearchLink" onclick="EnsureScriptFunc('Search.ClientControls.js', 'Srch.U', function()
{$find('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr').search($get('ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_sbox').value);})" href="javascript: {}"><img class="ms-srch-sb-searchImg" id="searchImg"
alt="Search" src="/_layouts/15/images/searchresultui.png?rev=23"></a><div class="ms-qSuggest-container ms-shadow" id="AutoCompContainer"><div id="ctl00_PlaceHolderSearchArea_SmallSearchInputBox1_csr_AutoCompList"></div></div></div>