To press the decline (or ignore) button on your cell phone thus rejecting an incomming phone call even when it's after 9, a weekend or you aint doing something THAT important.
Extra points if you're basil-ing some one who considers you a friend, your pal is in great need or you're doing it while the person is watching from a distance.
1)In da hood
guy1: yo what's hood son? did t-mack call you, he said you and him was gon' talk 'bout some bidness
guy2: yea nicca, he did call but i basil'd his ass
2)Somewhere in the fine county of England
guy1: Well good evening sir, fancy some tea?
guy2: yes indeed, mate, do you mind if I include Sir Taylor in our gathering?
guy1: No, but I reckon he will basil you