Here I have created a sample, which doesn't breaks the alignment of textbox with Bootstrap.
HTML
<div class="row">
<div class="col-md-2 text-right col-last">
Search By :
</div>
<div class="col-md-2 col-last">
<asp:DropDownList ID="ddlStatus" runat="server" CssClass="form-control">
<asp:ListItem Text="ID" Value="1" />
<asp:ListItem Text="Type" Value="2" />
<asp:ListItem Text="Status" Value="3" />
<asp:ListItem Text="Date" Value="4" />
</asp:DropDownList>
</div>
<div class="col-md-2 col-last">
<asp:TextBox ID="txtSearch" runat="server" placeholder="Search" CssClass="form-control"></asp:TextBox>
</div>
<div class="col-md-2">
<asp:LinkButton ID="btnSearch" runat="server" CssClass="btn btn-blue" Text="Search"></asp:LinkButton>
</div>
</div>
Screenshot
