Talking frankly people will put their trust to banks rather than with bitcoin. As we know banks are already operating for so many years and helping people with their services such as with loans, savings and other things that can help an individual for investment. There is no reason that we shouldn't trust banks, they are even reputable for they are supported by the government.
Bitcoin is young but the growth it has achieved is very high compared to any stock market other commodity. This clearly described that bitcoin is making revolution in the economic and banking system. Now upon the potential of bitcoin, banks have started to adopt the technology of bitcoin to be in the mainstream. So banks will be trusted by users even if bitcoin gets acceptance similar to banks.