Use a regular expression.
var dateRegEx = /^(0[1-9]|1[012]|[1-9])[- /.](0[1-9]|[12][0-9]|3[01]|[1-9])[- /.](19|20)\d\d$/
console.log("06/06/2012".match(dateRegEx) !== null) // true
console.log("6/6/2012".match(dateRegEx) !== null) // true
console.log("6/30/2012".match(dateRegEx) !== null) // true
console.log("30/06/2012".match(dateRegEx) !== null) // false
Learn about RegEx.
Edit - A Disclaimer
As @elclanrs pointed out, this only validates the string's format, and not the actual date, which means that dates like February 31st would pass. However, since the OP only asks "to validate date string format," I'll keep this answer here because for some, it might be all you need.
As a note, the jQuery Validation plugin that the OP was using, also only validates format.
Finally, for those wondering, if you needed to validate the date, and not just the format, this regular expression would have ~2% rate of failure over the domain of (1-12)/(1-31)/(1900-2099) date strings. Please don't use this in Mission Critical code for JPL.