The eligibility criteria to apply for Bank exams is Graduation. So you will be asked basic English vocabulary, grammar, comprehension etc in your exam. You will have to study topics like Active and passive voices, Spotting the error, multiple meanings, sentence completion, vocabulary, idioms and phrases, paragraph completion, reading comprehension, fill in the blanks, para jumbles and cloze test.Few books that can help you are:

• Comprehensive guide to SBI Bank PO by Disha Experts
• Bank PO English Language Solved Papers by Arihant Publishers
• Objective General English by S.P. Bakshi
• English for Bank Exams by S.Chand Publishers.

So if you are not fluent in speaking in English, you can watch Youtube channels in English speaking courses and read News Papers daily. Wren and Martin is also a goof book for you if want to study and improve English. You need to learn English for the exam as English secion contains 30 questions for 30 marks. Also there 0.25 deduction for every wrong answer. Cloze test, Reading comprehension, fill inthe blanks, paragraph or sentence correction will carry more marks and so your focus should be on these sections.

English is not a barrier. You can surely appear for the bank exams. To improve your English, before anything, first you need to communicate with the person sitting opposite to you. Unless you communicate with that person, you won't be able to discuss anything and describe yourself. That's why English is important. Not only English, your communication skill plays a great role here. Interviewers quickly make judgments and give great importance to first impressions. English gives you an advantage in terms of communication as you can reach more people of different linguistics.
Apart from this, you should follow the syllabus to prepare for Banking sectors. The syllabus of different banking exams are almost same, so you can prepare for different banking exams like - SBI PO, SBI Clerk, SBI SO, RBI Grade B, IBPS PO, IBPS Clerk, RBI Assistant exam etc. The syllabus usually consists of topics like Reasoning ability, Quantitative aptitude, Numerical Ability, Banking Awareness, English Language and Verbal Ability, Computer Awareness and Current affairs. First get the accurate syllabus. You must work on some numerical section to get a grip on them. Read banking awareness questions and answers. Read newspapers daily to keep the current affairs upto date.