I believe the cheapest plan offered right now is the streaming-only option for $7.99 per month. While I wouldn't sign up for Netflix just for the miles, it is a nice bonus if you already had plans to join in the near future. Allow 8-10 weeks for the miles to post to your account. You must link directly from this page and enter a valid American Airlines AAdvantage number to qualify for this offer.
EDIT: Here is an even better Netflix offer where you can earn up to 4,000 United MileagePlus miles after becoming a paid member. The terms and conditions are virtually identical to the offer above, except you will first earn 2,000 miles after signing up, and then you will another 2,000 miles if you pay with your MileagePlus Visa credit card. You can read more about the MileagePlus Visa here. You must link directly from this page and enter a valid MileagePlus number to qualify for this offer.