You can do
datePicker.setMinDate(System.currentTimeMillis() - 1000);
which sets today’s date as minimum date and all the past dates are disabled.
datePicker is an object of DatePicker if you are using an object of DatePickerDialog you can do
datePickerDialog.getDatePicker().setMinDate(System.currentTimeMillis() - 1000);
Note: setMinDate was introduced in API 11