Views: greater than a date expose filter
In Views, add a filter of Node: Post date.  You should see something like the image below (without example values changed):



Click Expose to have the filter exposed to your users, choose the operator Is greater than or equal to and choose how you want to filter the results (by inserting a date to be greater than or equal to or entering a value of "-6 months", etc).  This won't present a dropdown, but it will certainly add the filter you need for your view.