Quasi was originally a Latin word meaning as if, and it’s now an English word meaning seeming, seemingly, sort of, or in the nature of. It works as either an adjective or an adverb, and it’s frequently used in phrasal adjectives.
When quasi is a standalone adjective modifying a noun, no hyphen is required, yet by convention many writers do use a hyphen to affix quasi to the noun it modifies, creating a quasi-compound noun—for example:
[P]owerful multinationals have conducted themselves like quasi-states. [NY Times]
In rare cases, quasi may be similarly attached to a verb, becoming a quasi adverb—for example:
No matter how beautiful someone is, when you cheat, even quasi-cheat, you feel dirty. [New York Magazine]
The hyphen isn’t necessary in cases like these, but many writers are reluctant to treat quasi as a standalone word.
Quasi in phrasal adjectives
Quasi is most commonly used as an element of a phrasal adjective. In this case, quasi should be connected by a hyphen to the other part of the phrasal adjective—for example:
We’re going to do a college contest where schools will represent themselves in a series of quasi-athletic events. [Boston Globe]
To audiences starved for quasi-medical gore, the gruesome “Repo Men” should help fill the void. [NY Times]
The film’s logic has as much to do with architecture as narrative, a movement from personalized squats to quasi-middle-class uniformity. [LA Times]