Bitcoin collapsing the banking system is actually an impression that makes the government disapprove bitcoin. The idea of a cashless country, destroying their monetary unit is not what a country's government wants. The government will be at ease if the users are responsible enough and understanding enough to know the limits of a cryptocurrency in every country. From there, probably more countries will see potential and credit in bitcoin. Government is there to control and secure their citizens (especially from unstable currencies ...and probably scams and hacking) that's their job.
It is very unlikely that we are going to see bitcoin collapsing the banking system, banks will have a lot of time to do a lot of things, they could try to pass laws against bitcoin and the governments may agree on that on the grounds that bitcoin is a threat to the current economic system and the well being of their citizens or they may choose to adapt to this new circumstances and keep doing business as usual, after all banks do not have any ideology they only care about the money.