Consistently, my colleagues have made remarks along the lines of "I'll take the 1,000th flight after I know it's safe enough". I grew up with cars and airliners so I'm not sure what the process of convincing the public of BFR's safety will actually look like or how long it will take. For all I know, they'll give out the first 100 rides for free or at huge discounts, making all of these thoughts moot.
To put it another way, it seems likely that the safety will increase by that much. Before airplanes, hang gliders were one of the most popular ways to fly, along with balloons. Gliders were incredibly unsafe, because pilots were expected to control the glider by shifting their body weight. This made stalls fatal, since you couldn't recover.