Green’s Dictionary of Slang

1. (drugs) to take care of someone either under the influence of a drug (esp. the hallucinogenic LSD) or, more often, recovering from an unpleasant, drug-induced experience.

1975		G. Underwood ‘Razorback Sl.’ in AS L:1/2 55: babysit vt Stay with and attend to a person recovering from the bad effects of a drug.
2001		ONDCP Street Terms 2: Babysit — Guide someone through first drug experience.

2. to monitor progress, to take care of, to watch over; thus babysitting used as n. and attrib.

1970		(con. 1928) Gaddis & Long Panzram (2002) 126: The sergeant told Lesser that his job was to ‘babysit that dingbat.’.
1997		E. Little Another Day in Paradise 15: Danny boy here is gonna baby-sit you, make sure ya eat and rest.
2000		G. Pelecanos Shame the Devil 262: Detective Jonas is Boyle’s baby-sitting job, right?
2002		S. Maloney Something Fishy (2006) 130: And weren’t petty crims sometimes employed to baby-sit dope plantations?