1. Generate Web-to-Case HTML with the Status field

2. Remove the following code.

<label for="status">Status</label><select  id="status" name="status"><option value="">--None--</option><option value="New">New</option>
<option value="Working">Working</option>
<option value="Escalated">Escalated</option>
<option value="Closed">Closed</option>
</select><br>

3. Add the following code instead. I have used value as New from Web to Case. Please change it as per your org setup.

<input type="hidden"  id="status" name="status" value="New from Web to Case" /><br>

Output: