I went through this a few years ago, my trailer looks like the ones on this site:
Ratcheting tie-down straps are exactly what you want, you need three straps per bike. Put the bike in the middle rail. Hook one strap to each side of the handlebars, right where they curve from horizontal to vertical. Hook the other end to a point on the trailer as far out on the trailer as possible, you want the straps to extend out at an angle, not down.
Then, crank the straps down a little at a time on each side until the straps are tight and the front suspension is compressed a few inches. Make sure the bike is straight up and down.
These two straps do 90% of the work holding the bike down.
Then, put a third strap on the back of the bike--hook one end to the trailer, loop the strap over the rear rack or some other well-anchored part of the bike, hook the other end to the trailer on the other side of the bike, then crank that one down until it's tight.
The bike won't go anywhere. If you like, you can hook a bungee or something through the front wheel.
You'll love having a trailer, and once you have the strapping down thing figured out, it'll be easy.