1.Login to blogger dashboard-->Layout > Page Elements

2.Click on 'Add a Gadget'.

3.Select 'HTML/Javascript' and add the code given below and click save.

<script type="text/javascript">
var browser=navigator.appName;
var b_version=navigator.appVersion;
var version=parseFloat(b_version);

document.write("Browser name: "+ browser);
document.write("<br />");
document.write("Browser version: "+ version);
</script>

Now you are done.