Yes, identification cards would help stem illegal immigration because if one is asked to show ID and does not possess a card, then one would find it difficult to do business anywhere, or worse for them, may find themselves swiftly deported. Therefore, such ID cards would help stem illegal immigration.
Although ID cards seem like a good idea to help battle illegal immigration, it wouldn't be long before people learn to create fake IDs and the whole system becomes useless. Nowadays with the amazing amount of advanced technology at our disposal, it doesn't take much for someone to learn how to create a forgery of something like an ID card. For the cards to be successful they would have to be highly regulated, which would most likely be too large of a challenge in the long run.