First, be aware that the front-end Quick Search function may not work if you are viewing the store while logged into your Online Store Manager. This is due to the Admin Design bar that is applied to the "View Store" link when you are logged into your administration section. To test its functionality, please make sure to first log out of your online store manager or use a different browser than the one you're logged into the store admin with.
Next, check to make sure the quicksearch is not set to be hidden. Using the left navigation menu in the admin, go to Modules and search for the "Quick Search" module. Click on its + icon to expand it and ensure the Enable Quick Search box is checked.
(Advanced) If neither of the first two suggestions works, you may need to check some of your templates. This can be the case if you have customized templates that may have the coding removed from them. If this is the case, check for the following:
On your frame.html template
Make sure the following code is present. It should be placed prior to the closing head tag (</head>):
<link rel="stylesheet" href="assets/templates/common/quicksearch/quicksearch.css">
Also, within the frame.html template, check to make sure the search frame coding contains the following:
<form name="searchForm" method="get" action="search.asp">
<input name="keyword" id="searchlight" type="text">
<input type="submit" name="search" value="GO">