The badge is not causing the problem, you can check next example with two navbars (one with badge filled with a number, other without it) where both have similar height (like Chris Bar already noted). So:
1) It could be a problem with some of your custom CSS, in this case, I recommend to use the inspector tool to check what style is added to your badge or navbar.
2) Another thing you could do is check if you have a well structured navbar as explained on Bootstrap Navbar.
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<!-- Latest compiled and minified CSS -->
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" integrity="sha384-BVYiiSIFeK1dGmJRAkycuHAHRg32OmUcww7on3RYdg4Va+PmSTsz/K68vbdEjh4u" crossorigin="anonymous">
<!-- Optional theme -->
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap-theme.min.css" integrity="sha384-rHyoN1iRsVXV4nD0JutlnGaslCJuC7uwjduW9SVrLvRYooPp2bWYgmgJQIXwl/Sp" crossorigin="anonymous">
<!-- Latest compiled and minified JavaScript -->
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha384-Tc5IQib027qvyjSMfHjOMaLkfuWVxZxUPnCJA7l2mCWNIpG9mGCD8wGNIcPD7Txa" crossorigin="anonymous"></script>
<nav class="navbar navbar-inverse navbar-fixed-top">
<div class="container">
<!-- Brand and collapse button -->
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar1">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Navbar 1</a>
</div>
<div id="navbar1" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li id="vugraph-menu" class="nav-item dropdown" title="No tables in play">
<a id="navbar-vugraph-menu-link" class="nav-link dropdown-toggle" href="#" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">
Vugraph
<span id="vugraph-count" class="no-padding badge badge-notify">99</span>
</a>
<ul class="dropdown-menu" aria-labelledby="navbar-vugraph-menu-link">
<li class="dropdown-item">
<a href="https://www.bridgebase.com/v3/?vugraph=y" target="_blank">Watch</a>
</li>
<li class="dropdown-item">
<a href="https://www.bridgebase.com/vugraph/schedule.php" target="_blank">Schedule</a>
</li>
</ul>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/myhands/index.php" target="_blank">Hand Records</a>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/points/index.php" target="_blank">Masterpoints</a>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/purchase/pay.php" target="_blank">BB$</a>
</li>
</ul>
</div>
</div>
</nav>
<nav class="navbar navbar-inverse navbar-fixed-bottom">
<div class="container">
<!-- Brand and collapse button -->
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar2">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Navbar 2</a>
</div>
<div id="navbar2" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li id="vugraph-menu" class="nav-item dropdown" title="No tables in play">
<a id="navbar-vugraph-menu-link" class="nav-link dropdown-toggle" href="#" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">
Vugraph
<span id="vugraph-count" class="no-padding badge badge-notify"></span>
</a>
<ul class="dropdown-menu" aria-labelledby="navbar-vugraph-menu-link">
<li class="dropdown-item">
<a href="https://www.bridgebase.com/v3/?vugraph=y" target="_blank">Watch</a>
</li>
<li class="dropdown-item">
<a href="https://www.bridgebase.com/vugraph/schedule.php" target="_blank">Schedule</a>
</li>
</ul>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/myhands/index.php" target="_blank">Hand Records</a>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/points/index.php" target="_blank">Masterpoints</a>
</li>
<li class="nav-item">
<a class="nav-link" href="https://www.bridgebase.com/purchase/pay.php" target="_blank">BB$</a>
</li>
</ul>
</div>
</div>
</nav>Finally, I don’t believe that updating to Bootstrap 4 will help here, indeed, you will have to port your entire site to the new version because there are thing that changed from version 3 to 4 (Read more here).
Update
After the question was updated with the custom style applied on the elements, a possible solution is to add next style to the badge:
{line-height: 0; display: inline;}
Example:
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/1.11.3/jquery.min.js"></script>
<link href="https://stackpath.bootstrapcdn.com/bootstrap/3.3.6/css/bootstrap.min.css" rel="stylesheet">
<style type="text/css">
/* Move down content because we have a fixed navbar that is 50px tall */
#vugraph-count {
line-height: 0;
display: inline;
}
body {
padding-top: 33px;
padding-bottom: 20px;
}
.body {
font-size: 16px;
border-spacing: 2px;
border-collapse: separate;
text-align: center;
}
.body a {
text-decoration: underline;
}
.navbar-fixed-top {
background-color: #5b5b5b;
min-height: 33px;
border-style: none;
}
.navbar-toggle {
margin-top: 2px;
margin-bottom: 2px;
padding-top: 7px;
padding-bottom: 7px;
}
.navbar-nav>li>a {
line-height: 3px;
padding-top: 16px;
padding-bottom: 14px;
}
div.navbar-header>img {
height: 30px;
margin-top: 2px;
}
.nav>li>a {
font-size: medium;
font-family: "Helvetica Bold", sans-serif;
/*font-weight: bold;*/
text-transform: uppercase;
}
.navbar-inverse .navbar-nav>li>a {
color: #f0eed0;
}
.navbar-inverse .navbar-nav>li>a:hover {
color: #f8dc54;
}
.nav li {
display: inline;
}
#vugraph-menu.inprogress>a {
color: black;
background-color: #ecbe23;
}
#vugraph-menu.inprogress>a:hover,
#vugraph-menu.inprogress.open>a:not(:hover) {
color: white;
}
</style>
<script src="https://stackpath.bootstrapcdn.com/bootstrap/3.3.6/js/bootstrap.min.js"></script>
<nav class="navbar navbar-inverse navbar-fixed-top">
<div class="container">
<div class="navbar-header">
<img src="http://www.bridgebase.com/images/bbo_red.png">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false" aria-controls="navbar">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
</div>
<div id="navbar" class="nav navbar-collapse collapse">
<div class="container">
<ul class="nav navbar-nav">
<li id="vugraph-menu" class="nav-item dropdown" title="No tables in play">
<a id="navbar-vugraph-menu-link" class="nav-link dropdown-toggle" href="#" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">Vugraph <span id="vugraph-count" class="badge badge-notify">1</span></a>
<ul class="dropdown-menu" aria-labelledby="navbar-vugraph-menu-link">
<li class="dropdown-item"><a href="https://www.bridgebase.com/v3/?vugraph=y" target="_blank">Watch</a></li>
<li class="dropdown-item"><a href="https://www.bridgebase.com/vugraph/schedule.php" target="_blank">Schedule</a></li>
</ul>
</li>
<li class="nav-item"><a class="nav-link" href="https://www.bridgebase.com/myhands/index.php" target="_blank">Hand Records</a></li>
<li class="nav-item"><a class="nav-link" href="https://www.bridgebase.com/points/index.php" target="_blank">Masterpoints</a></li>
<li class="nav-item"><a class="nav-link" href="https://www.bridgebase.com/purchase/pay.php" target="_blank">BB$</a></li>
</ul>
</div>
</div>
</div>
</nav>9
solved Badge increases navbar height