No you can not apply for the Jobs in BSNL after passing 12th. to apply for the BSNL the applicant should have completed graduation with good academic record from any recognized university of India.

Other details BSNL recruitment examination:
Age limit
21 - 30 years.

The selection procedure of BSNL is as following:

WRITTEN TEST.
PERSONAL INTERVIEW & CERTIFICATE VERIFICATION.

Available posts at BSNL:
Junior telecom officer – JTO
Technical telecom assistant – TTA
Database management
Networking engineer
Programmer
Telecommunication engineer
Accounts engineer etc