There are some marketplace widgets available that can do this – the month and year picker comes to mind, for instance. I don't have personal experience with them, so you'll have to see if it does everything you like.
You can use the customizable Datepicker from the marketplace module,
there you can set the attribute and minimum date as today and all the dates before today will get unclickable.
Let me know if you face any issues.
Hope it helps!!