Some good solutions were presented in an ALSC post, but none of them quite worked for our library. I didn't want to alphabetize the books since that would constantly be a losing battle, and our room has too much traffic to try the sandbox approach. Companies that make board book shelving also want about a million dollars for each unit, which definitely isn't helpful. So, I searched and searched until I finally came up with a solution.
This is working beautifully. Our board books are flying off the shelves and caregivers have loved the new shelving. Right now, 75% OF THE COLLECTION IS CHECKED OUT!
We also made the decision to make our board books non-holdable. Nine times out of 10, people were just putting specific books on hold without realizing they were choosing a board book. Making this decision has made our holds lists so much easier in the morning, and I highly recommend it. What are you using for board book shelving? Let me know!