I took up bouldering around 2 years ago, despite my fear of heights. I've managed it by sticking exclusively to gyms, and staying well within my limits. What also helps is the gym I started at had a safety orientation that all climbers had to go through before they'd be allowed on the walls. The most important part of this was how to fall correctly.

The gym I frequent also has a great variety of route difficulties and heights (~3m to ~7m), so it really helped me move up slowly along with my comfort level. As for the amount of times I fall, it's far less than I expected, and I've never felt unsafe, even while falling.