It's been my experience that table service restaurants are more likely to not ask to see the blue card than retail venues. Only twice has someone asked to see my id to check that it matched the blue card. Once was when activating an AP. The second was at a DS dining location that was offering a 40% off DVC promotion for lunch.
Note that I am not saying that table service restaurants and retail establishments don't ask for the card. Most of the time they do. But, I've had cases where they took my word that I was eligible for a DVC discount, or scanned my band and saw I was at BLT or BCV and just subtracted the discount without even asking. This was quite common five years ago, and I feel that Disney's training program has stepped up and made the practice relatively uncommon today.
Plan on carrying your blue card and your id. Despite the website saying you can print your own card and present that, don't believe it. If you don't have your physical card, the CM will more than likely accept you showing it to them on your phone (although I did have one CM reject that until their manager stepped in).