I'm partial to the WIKE trailers, I have two of them that I use to haul my kids. They make a line of cargo trailers that start around $100 (see http://www.wicycle.com/cargo.htm) . I think even the small model would carry a guitar well.

Based on my experience hauling kids I would say the guitar will be pretty well protected against breaking. I have crashed my bike and done all sorts of stupid stuff like riding into curbs without tipping the trailer. What I would be most worried about is vibration, which will cause your guitar to rattle to pieces. Just about everything on a bike uses locktite or locknuts to keep from falling apart, but not on a guitar. I would put the guitar in a road case, or on a piece of foam to isolate it from the trailer.

I think I would experiment with a less valuable instrument before strapping up my #1 axe as well.

Nick