Etymonline is generally considered to be the more verbose and reliable reference for etymologies. Its entry for basilisk reads:
c.1300, from Latin basiliscus, from Greek basiliskos "little king," dim. of basileus "king" (see Basil); said by Pliny to have been so called because of a crest or spot on its head resembling a crown.
Basilica's entry reads:
1540s, from Latin basilica "building of a court of justice," and, by extension, church built on the plan of one, from Greek (stoa) basilike "royal (portal)," the portico of the archon basileus, the official who dispensed justice in Athens, from basileus "king" (see Basil). In Rome, applied specifically to the seven principal churches founded by Constantine.
Wikipedia's entries for basilisk and basilica confirm the royal connection that the two words share. The former includes an excerpt from the aforementioned Pliny's encyclopaedia published in ~79CE:
There is the same power also in the serpent called the basilisk. It is produced in the province of Cyrene, being not more than twelve fingers in length. It has a white spot on the head, strongly resembling a sort of a diadem.