The source for the custom of spitting during the prayer of Aleinu is the Turei Zahav, Yoreh Deah 179, Siman 5. The reason for the custom is in order not to benefit from the spittle that forms in one’s mouth while mentioning idol worship. (HaYomYom 9th of Tevet.) The solution to the problem of causing discomfort in your environment is to swallow your spittle just before you say “And they bow down” (SheHem mishtachavim lehevel…) and then you won’t have it in your mouth when you say those words. In this manner you won’t be disregarding the custom, and you also won’t cause unpleasant incidents.
