Yes, there is a train station at Larne port which is for local use - Larne Harbour Train Station. This service runs in connection with the ferries. There is also a main station in Larne which is a 15 minute walk away from the port and connects to Ireland's major cities.
Larne bus station is located in the town center, which is a 20 minute walk away from the port, unfortunately there is no direct bus link to the port.