A major police response was triggered when the man went into branches of Santander and Barclays in King Edward Street with a container of what is believed to be petrol and tried to start fires.
Firefighters put out a blaze in one of the branches.
A suspect was arrested at the Paragon Interchange bus and rail station.
Humberside Police has said the incident is not being treated as terrorism and the "situation was contained very quickly by police and emergency crews".
Chief Superintendent Phill Ward, North Bank Commander, said: “We are mindful of the public's concern that these types of incidents may be related to terrorism, however there is nothing to suggest that this is anything other than an isolated incident which has been dealt with promptly by the responding police and emergency services.
“I want to offer reassurance that our priority at all times is the safety of the public.”
Bank staff say they are co-operating with the police to work out the motivation behind the attack.
A Santander spokeswoman said: “Santander is shocked by the attack on its branch in Hull and we are cooperating with the authorities to understand what happened and why.
“None of our colleagues or customers were physically injured in the incident, and our immediate priority is to provide support and counselling to those who are understandably upset.”
A Barclays spokesperson said: "We can confirm an incident took place this afternoon in the Barclays Hull city centre branch.
"There are no physical injuries to staff, however, we are cooperating fully with the police and the branch will be closed until further notice."