stare (v.) Look up stare at
Old English starian "to stare, gaze, look fixedly at," from Proto-Germanic *staren "be rigid" (cognates: Old Norse stara, Middle Low German and Middle Dutch staren, Old High German staren, German starren "to stare at;" German starren "to stiffen," starr "stiff;" Old Norse storr "proud;" Old High German storren "to stand out, project;" Gothic andstaurran "to be obstinate"), from PIE root *ster- (1) "strong, firm, stiff, rigid" (see stereo- and compare torpor).

Not originally implying rudeness. To stare (someone) down is from 1848. Related: Stared; staring.
stare (n.2) Look up stare at
"starling," from Old English (see starling).
stare (n.1) Look up stare at
late 14c., "power of sight," from stare (v.). From c. 1700 as "a fixed gaze."