From Dutch baas, from Middle Dutch baes, from Old Dutch *baso, from Proto-Germanic *baswô, masculine form of Proto-Germanic *baswǭ. Cognate with Middle Low German bās ("supervisor, foreman"), Old Frisian bas ("master"), hence Saterland Frisian Boas, Old High German basa ("father's sister, cousin"), hence German Base.
Originally a term of respect used to address an older relative. Later, in New Amsterdam, it began to mean a person in charge who is not a master. The representation of Dutch -aa- by English -o- is due to the older unrounded pronunciation of this letter, which is still used in North America and parts of Ireland, but was formerly found in some British accents as well.
The video game sense is borrowed from .
From Middle English bos, bose, boce, from Old French boce, from Frankish *bottja, from Proto-Germanic *bautaną. beat; see there for more.