stareyoudaoicibaDictYouDict[stare 词源字典]
stare: [OE] The etymological notion underlying stare is of ‘fixity’ or ‘rigidity’. It goes back ultimately to the prehistoric Germanic base *star-, *ster- ‘be rigid’, which also produced English starch [15], stark [OE], starve (originally ‘be stiff’, hence ‘die’), stern ‘severe’, and stork (etymologically the ‘stiff’-legged bird). Thus to stare is to ‘look fixedly’. (Greek stereós ‘solid’, source of English stereo, came from the same Indo-European base as produced *ster-.)
=> starch, stark, starve, stereo, stern, stork[stare etymology, stare origin, 英语词源]
stare (v.)youdaoicibaDictYouDict
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)youdaoicibaDictYouDict
"starling," from Old English (see starling).
stare (n.1)youdaoicibaDictYouDict
late 14c., "power of sight," from stare (v.). From c. 1700 as "a fixed gaze."