The old Julia Street cruise terminal parking lot is all torn up right now due to construction. You can park at several nearby private parking lots for around $12.00 per day. Most offer complimentary shuttle service to the cruise terminal.
Rob--if you plan to drive in a day early there is a hotel in New Orleans called the Maison St. Charles--they have really good rates on their cruise package with $5 per night parking (i think) and free limo to cruise terminal. there are also a few other perks.
The best and nearest place is the fulton place garage, it's a several story covered parking deck only a couple of blocks from the port. I parked there last week and had no problems, they have a free shuttle to and from the pier. The price is $12.00 per day. Hope that helps.