Err- define "heights".
The deck of a frigate to the waterline is about 20-30 feet. It is possible at some point in your career you may have to jump that far, but if that happens, I seriously doubt that your fear of heights will be the first thing on your mind
A serious answer to your question is probably in the form of several questions:
1) Are you going to be a fire contol, radar, or naval weapons tech? If so, you may spend significant portions of your day "aloft" fixing antennas or working on the CIWS (or so I have observed). These activities can see you working anywhere from 50-170 feet above the water- well strapped in, I might add.
2) If you are going to be a bosun, you could find yourself painting almost anywhere I have just described above in 1).
3) Anyone on a Ship could have occasion to be required to climb the jumping ladder from a small boat to the deck of the ship. That is about 25 feet straight up. And usually everything is moving...
I know several Sea King Navigators who are terrified of heights, yet have no problem hanging out in the open door of the helicopter of being hoisted from great heights. The "fear thing" seems situational...
The hard navy trades on the board may wish to add to or correct any of my statements above.
Cheers