First step would be the IMU CET. Check here for details It is conducted by the Indian Maritime University and has 6 campuses. Its a central University and exclusive for marine engineering.  The IMU has a list of colleges that are affiliated to it.

Apart from that, you could write CUSAT CAT for admissions to the Kunjali Marakkar School of Marine Engineering Faculty of Engineering. Check here for details

 VELS University offers B.E in marine engineering. The eligibility doesn't specify any entrance exam though the university has its own called VEE.

You could apply for COMEDK UGET and get admissions in the following colleges

• Srinivas Institute of Technology, Mangalore
• Mangalore Marine College and Technology, Mangalore

Check details here

If you belong to Tamilnadu, you could apply through TNEA to the following institutes

• SAMS COLLEGE OF ENGINEERING AND TECHNOLOGY
• SRI VENKATESWARA COLLEGE OF ENGINEERING (AUTONOMOUS
• G.K.M. COLLEGE OF ENGINEERING AND TECHNOLOGY
• SRI NANDHANAM COLLEGE OF ENGINEERING AND TECHNOLOGY
• PSN COLLEGE OF ENGINEERING AND TECHNOLOGY
• MOHAMED SATHAK ENGINEERING COLLEGE

You can check the top marine engineering colleges below

Amity University, Noida

Vels University, Chennai

Chandigarh College of Engineering and Technology, Chandigarh

CV Raman College of Engineering, Bhubaneswar

Indian Maritime University, Chennai

In order to take admission in these colleges you have to appear for engineering entrance exam like JEE Main and you can also take admission in your state level entrance exam