Guys, you are forgetting he is moving IN THE WINTER!!! If you are going to use a open trailer, the bike will need to be covered. All the Salt, Sand, road crap will get everywhere and will not be good for the bike. It is not like using a trailer in the summer with just rain. The sand used on roads in the west will just blast and pit the plastic.
I agree that a motorcycle cover or tarp will do major damage.
Here is what I did when I moved my bikes from Penn to Washington State in Feb. on an open snowmobile trailer. I went to a moving company and picked up a two foot wide roll of the shrink wrap they use to wrap furniture. Put the bike on the trailer, tie it down, then wrap the bike in the shrink wrap. I then used packing tape and wraped that all over the shrink wrap. Just make sure you also get the wheels covered. I did not get them covered on one bike and the cast rims had pits in them from the salt. Also, take off the wind shield and side bags if you can. Just less to wrap if you do it this way. After 3000 miles, took off the wrap, and bikes were fine.
The best thing like others have posted is to get a covered trailer.