What is their problem with the G20 meetings?
Losing national sovereignty to world political bodies that circumvent individual country's political systems and constitutions by setting policy that directly affects the lives of citizens without being directly accountable (ie:voting).
That is my personal problem with these meetings. The EU is an example of countries losing national sovereignty in this manner. It starts out with trade agreements like NAFTA and progresses from there.
The punks that get the media attention you speak of are generally provocateurs. They are planted to give a rational for the excessive force the police exercise on protesters.
That is my opinion though. Honestly, if the protesters want to express their opinion, they should be allowed to irrespective of the validity of their arguments or what they want to accomplish. Basic human rights are being violated, and everyone is debating on whether the protester's viewpoint is credible.