Hi Ruchika,
If you wish to persist DOB for the user then you need to store DOB in profile based on your mbox settings it will last for 30 days or more and you can use it on different page.
while creating mbox or updating mbox you can use below syntax to store it in profile.
<script>
mboxCreate('mboxName','mboxName','profile.DOB=06/23/2017','param2=value2');
</script>
or
<script>
mboxDefine('someId','mboxName','profile.DOB=06/23/2017','param2=value2');
mboxUpdate('mboxName','param3=value3','param4=value4');
</script>
once it's stored in profile at your home page offer code can read it with below code
<script>
mboxFactoryDefault.addOnLoad(function(){
//use strDOB variable and set it in banner.
var strDOB = "${profile.DOB}";
console.log("skus" +strDOB);
});
</script>