Okay, this is according to Hanks & Hodges, so don't blame me for the meaning of this surname...
Kwasinski is probably a variation of the surnames Kwasniewski and Kwasniak. It's a Polish nickname "for a sour-faced individual" ("kwasniak"), from the Polish word "kwasny" meaning "sour, peevish", which comes from the Polish word "kwas" meaning "acid".
-- Nanaea