Answer:
It is more common to not have one.
Not true: Most people have some sort of mark on their bodies at birth, but most fade/ disappear totally in the early months/ years of life. Many people have faint pink birthmarks on their heads or back of the neck ("stork marks") which generally are covered by their hair, and are, therefore, not very noticeable.
The majority of Asian babies (probably 70%+), and often those with some Asian ancestry, are born with blue or purple marks on the base of the spine or the buttocks (the so-called "Mongolian mark"), most of which fade/ disappear early in life... but not always. My wife has a large blue Mongolian Mark at the base of her spine.
It is true that "classic" brown, black or red/ purple birthmarks are less common: many medical authorities estimate about 10% of most populations bear such marks. However, I'm not so sure about this figure. I have a brown birthmark on one of my buttocks, and used to be very self-conscious about it, thinking it "freakish": until I spent some time at a nudist beach, and couldn't help but notice that a lot more than 1 in 10 people - probably 30% (especially women) - had marks like mine. I reckon the incidence of such marks is under reported, and in a very image conscious society, many people go to great lengths to disguise such birthmarks.
Finally, most people have at least 20 smaller marks ("moles") on their bodies, although most of these emerge later in life (often at puberty), and, therefore, are not technically "birthmarks".
IMO, most people do have some sort of birthmark; many, however, either fade rapidly in early life or are not very visible. Also, many people are very self-conscious about their birthmarks, and spend huge amounts of time and energy covering them up.