I do not know if show has shown this or not but as per books Arthur Dayne was present at Tourney of Harrenhal (That's the Tourney where Rhaegar met Lyanna and later ran away with her with help of none other than his close friend Arthur Dayne). Eddard Stark was also present there. So it is plausible that Dayne met the younger son of Lord Rickard Stark, Warden of the North, second in line to seat of the Kings of Winter after his elder brother Brandon.
Even if they did not meet, it is most likely that the Starks being nobles of a great house were granted seats in the VIP Pavilion of Lord Whent & King himself. It is a tradition that jousters turn to salute the host of the tourney and King (If present). Arthur rode in the lists at the tourney and thus he must have saluted the host, where he might have seen the "Shy Wolf" Eddard Stark.
Not to mention, Ashara Dayne (Sister of Arthur Dayne) danced with Eddard Stark at Harrenhal and supposedly she and Eddard fell in love with each other. Being a dutiful brother, Arthur might have been keeping an eye out for his sister (Seeing which of the young lordlings and knights were hitting on his sister. Fun fact, his own sworn brother Barristan Selmy was also secretly in love with Ashara) as well, thus getting to know Eddard Stark.
In conclusion, it is hard to assume that Arthur Dayne was not familiar with Eddard Stark.