Assuming these are nice records, and depending on where they're going for storage, you may want to consider something more protective than cardboard boxes. In moving my collection across the country, I once used those big Rubbermaid bins (from Target/HomeDept/Staples) - which hold about two boxes worth of records - and then sealed them with duct tape. Obviously these will make heavier packages (i.e. harder to move around) but they will protect against water damage or mold from other things in storage. And you can also stack other things on top of them without damaging the records.
If you're going to use cardboard boxes, make sure they're off the ground (again, re: flooding). The nice thing about plastic crates is that the newer ones usually have feet to lift them off the ground by an inch or so.