Is what you are taking off milk or colostrum? If the latter I would definitely freeze it (I am not sure when they start to make colostrum, never had to milk out before kidding).
Also if she is dripping milk, you need to make sure that she doesn't get any muck (or bugs) into the teat canal - so I would definitely teat-dip with an iodine solution, or just spray the teat with something like bactakil maybe?