(This is a paraphrased version of something I learned from BCnrd. Consequently I'm posting it as a CW answer.)

There is currently no known useful criterion for quasi-compactness from the functor of points. Nonetheless "locally of finite presentation" does have one (cf. EGA IV-8.14).

This means in particular that to check for properness via the valuative criterion (which is purely functorial, of course), the extra steps of ensuring finite type need additional work---locally of finite type can be checked thanks to the criterion just listed, but one also wants quasi-compactness.

So in practice what one often does is to take something known to be quasi-compact (e.g. a concrete scheme) and surject this onto the moduli space in question. Surjectiveness can be checked functorially (using the spectra of large algebraically closed fields). This is the main way of establishing that an Artin stack or algebraic space is quasi-compact.