Fencer: You trained with blades for long hours as a youth, either taking lessons in the genteel art of fencing from tutors paid for by your parents or being taken under the wing of a disenfranchised fencer who may have turned to a life of crime. You gain a +1 trait bonus on attack of opportunity attack rolls made with daggers, swords, and similar bladed weapons.
(Daggers), (Swords), and (similar bladed weapons). That's three distinct things that Fencer works with. So even a non-bladed dagger or sword would count and a Rapier is most certainly a type of sword. Thus, the point of whether a rapier is inherently bladed is moot as it would qualify whether or not it's bladed.
For those still interested, read on. When most people think of a Rapier, they probably think of a fencing foil or epee used in fencing as a sport; these don't have sharpened blades along the length (though, the length of the weapon is still referred to as the "blade"). Rapier is a German term used to identify what was, from the perspective of the Germanic people, a foreign weapon; the same weapon was referred to as Spada, Espada, and Epee in Italian, Spanish, and French respectively. It was noted as being lighter than a side-sword (used by armored knights) but heavier than a smallsword (used by cavalry officers). The edges tended not to be sharpened but, again, the "business end" of a Rapier is still referred to as a "blade".
The origin of the Rapier goes back to 1500s Spain and the espada ropera (dress sword). It was a "cut-and-thrust" style weapon and actually did bear sharp edges though, with a thin blade and less weight behind it, "cutting" style attacks were very weak compared to more robust swords. Later fencing masters tended to favor pure thrusting styles, though, there were still a few "die-hard" fans of cut-and-thrust techniques.
While rapiers tended to be a more "civilian" weapon, for personal defense, more "combat emphasized" versions were developed. The colichemarde used a standard rapier hilt but with a much heavier blade which was better at parrying and for more robust combat. It is sometimes referred to as a "transition rapier". This, in turn, gave way to the "smallsword", which used a much more simple hilt and was less cumbersome to carry around and highly suited as a melee sidearm for the battlefield. There were also "war rapiers" which had a wide blade, but on a rapier-like hilt.
In general, rapiers usually had sharpened edges. At least prior to turning into silly playthings and accessories in later eras.
It is because you don't want that awkward situation you get with spears, where the guy grabs the thing you are stabbing him with. With edges, a quick snag would actually cause him quiet a bit of pain, and then you can stab him during that moment.