All vehicles will rust underneath in a very short amount of time. Heck, I bet some cars do this at the factory.
You said "underneath", but where exactly underneath? If it's on the floor pans, then that would be a concern. However a body shop can remove the old rust and put more undercoating underneath. I would suggest this due to your location.
If the rust is on suspension components, such as springs, shocks, tie rod ends, etc, then there is no reason to worry, as all of these parts will rust. My almost 20 year old car had a rusty suspension, but no kind of issue where safety would be of any concern.