Yes, a B.E student you can apply for the bank job, the basic requirement to apply for the bank job is that the applicant should have completed graduation; you are eligibility for following bank jobs:

Probationary officer
Clerical grade

Institute of Banking Personnel Selection conducts entrance examination for the recruitment of clerk and the Probationary officer. Following are the details of it:

Eligibility for clerk
The applicant should hold a degree in any discipline from a recognized University or any equivalent qualification recognized as such by the Central Government.

Age:
Minimum age 20 and Maximum age 28


Eligibility for PO:
The applicant should hold a degree in any discipline from a recognized University or any equivalent qualification recognized as such by the Central Government.

Age
Age requirement in most Bank PO exam is 21 to 30 years. This may be relaxed upto 5 years for reserved categories.

Nationality / Citizenship for Both:
A candidate must be either:
A Citizen of India or
A subject of Nepal or
A subject of Bhutan or
A Tibetan Refugee who came over to India before 1st January 1962 with the intention of permanently settling in India or
A person of Indian origin who has migrated from Pakistan, Burma, the United Republic of Tanzan Sri Lanka, East African countries of Kenya, Uganda, ia (formerly Tanganyika and Zanzibar), Zambia, Malawi, Zaire