I bought a pallet of boxes double walled from Bags Unlimited to have about 5,000 records moved across the country. I packed them myself, made sure that the riders on my insurance were up to date value (a blanket for the records) and the movers handled them fine. They went into a high value vault with HVAC at the mover's facility until we bought a house in our new location.
I subsequently found a better box, cheaper, on Amazon- same deal- double walled, sturdy.
Keep in mind that the insurance attached to the standard mover's agreement is pretty much worthless. (Not that insurance would make it easy to replace some of these things but it is certainly better than a loss).