stare (n.2) Look up stare at
"starling," from Old English (see starling).
stare (v.) Look up stare at
Old English starian "to stare, gaze, look fixedly at," from Proto-Germanic *staren "be rigid" (source also of 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.1) Look up stare at
late 14c., "power of sight," from stare (v.). From c. 1700 as "a fixed gaze."