This is not true, certainly for the later period and probably for the earlier period too.
Covering the early period (and bearing in mind that it is heroic fiction), we have references in Beowulf to the 'grimhelmas' worn by the warriors of Beowulf's company on arrival at Heorot (line 334)
and before the fight with Grendel (line 1245); none of whom were kings. While this isn't really strong evidence that helmets were worn regularly by warriors, it is a strong indication that there was no cultural reason stopping them from doing so.
Later, in the 11th century, we have good written evidence of helmet wearing in an edict from 1008 that requires every eight hides of land to provide a helmet and mail coat, written account of a gift from Earl Godwine to Harthacanute of a ship manned by eighty warriors 'of whom each one had on each arm a golden arm-ring weighing sixteen ounces, a triple corselet, on the head a helmet in part overlaid with gold;', and of course the Bayeux Tapestry, which depicts the majority of the Anglo-Saxon warriors wearing helmets.
Reasons for thinking only kings wore helmets
While there doesn't seem to be any real evidence that only kings wore helmets in this period, what we do have is a paucity of evidence of helmet wearing, and of the six relatively complete Anglo-Saxon helmets we do have, two are almost certainly of kingly origin (the Staffordshire hoard helm and the Sutton Hoo helm), and the others are from high-status graves (of note is the fact that the Coppergate helm is inscribed with the name 'Oshere', which is almost certainly not the name of a king).
It's probably true that in the earlier Anglo-Saxon period few people apart from kings could afford helmets, this doesn't imply that helmet wearing was ever restricted solely to kings.