BBA is a professional course which covers concepts of management and business administration and emphasises development of required skills. BCom, a bachelor degree, imparts knowledge related to finance and accounting alongwith various aspects of business and commerce. In a way, both courses are helpful for banking field.
You will be eligible for banking exam after doing BBA or BCom. In either case, you can also do MBA.
You can go for MCom, CA and other account and finance related courses after BCom. You can opt MBA - Finance and other branches for specialisation.
Doing BBA can be helpful for completing MBA which is one of the few choices after it. You can choose specialisations like sales, marketing, HR, etc.
MBA and CA can enable you for specialist officer jobs too. Banks sometimes hire from premier business schools, engineering, accounts, and other colleges directly.
If you like doing financial computations, better join BCom(hons). Thereafter, you can directly apply for junior management trainee posts in Banks. After CA, CS, CFA or MCom, you may even be eligible for middle management posts.
If you like management and administration field more and not much interested in finance and accounting, choose BBA.