Short answer: YES. According to a study, 59% of men 18-24 are terrified of going bald.
In an article titled Most Young People are "terrified" of going bald by yougov.com the author quotes a study which asked if people agreed with the statement "going bald terrifies me." If stated like that, almost anyone would agree with it, no?
In any event, for most men (and some women), this is a real concern. Especially for men in their 20's. The first signs of balding can indicate that you're no longer a kid. This is a tough reality from a vanity and overall life standpoint.
One section of the article worth exploring is:
"Male pattern hair loss is not common for most men under 30 years old, which perhaps explains the terror around going bald at a young age. As men get older, their likelihood of experiencing hair loss increases, and their concern appears to fade as it becomes more expected."
I don't entirely disagree with her. However, I think a more accurate statement would be: as men age AND the likelihood of hair loss increases, their concern fades because they have matured to a point where they can accept (and even embrace) their baldness. No longer attached to the fiction that they are the same as when they were a teenager, they can move on to being a man.
The author is a data journalist so her assessment is based on the data. So it makes sense that she wouldn't speak about maturity...but it would be an easy inference.
On another note, I was surprised at some of the data referenced. In particular, I didn't know how many women had a fear of thinning or balding. It may be more difficult for a woman to deal with losing hair than a man since for men, it's common and accepted. I've seen women who've lost their hair (due to alopecia or chemo therapy) or decided to shave it off (for whatever reasons) who look beautiful still. In some cases, their beauty comes out more without hair.