The answer is "maybe." According to our POS renters, even if they are members, are not entitled to member perks. In practice, however, it is likely you will get free internet, if your room key says "DVC Member" on it.
We're not DVC members, but we just returned from 9 days at BCV on rented points. I only used the internet one day as I was not willing to pay the $10 per day charge, however in reviewing our final bill, we were not charged for the internet at all. (Probably a good thing I did not know this at the time...!)
We also received the DVC 10% discount two of the four times we used the kids club. (DD begged to go every night!) I was only asked one time if we were DVC or not.
As Chuck mentioned, our key's did say "DVC member" on them.