A government shutdown won't be a huge deal. It will have many bad effects, but a brief shutdown has little lasting effect.
Hitting the debt ceiling, on the other hand, is a far more dangerous situation. I don't buy the argument that hitting the debt ceiling automatically means defaulting on our debt (which is why you hear politicians use weasel phrases like "defaulting on our obligations"), but it would be worse than a shutdown, largely because it's a giant unprecedented disturbance with something very complicated (kind of like Obamacare). We conservatives don't like large unknown disturbances with complicated systems. Annie Lowrey at the New York Times lays out that same argument.
But here's the twisted thing: politically, messing with the debt ceiling would probably be more popular than messing with a government shutdown.
Why? Because the stuff above is a bit opaque and complicated. "Government shutdown" sounds like "government stops working." It sounds like "cops don't show up to work," and "I don't get my Social Security checks."
On the other hand, "raising the debt ceiling," sounds like, well, in the words of a former U.S. Senator:
sign of leadership failure. It is a sign that the U.S. Government can’t pay its own bills. It is a sign that we now depend on ongoing financial assistance from foreign countries to finance our Government’s reckless fiscal policies. … Increasing America’s debt weakens us domestically and internationally.
So, if Boehner and Reid take any heat for fighting over a government shutdown, that's objectively more responsible than fighting over a debt-limit hike -- but a debt limit fight would be more politically popular.