The classical (pre Deligne-Mumford) approach is to map Mg into Ag using the Torreli map, and observe that A_g itself is a quasi-projective variety.
Reference: Mumfords curves and their Jacobians (now bundled together with the red book in LNM 1358) Lecture IV.
In Lecture II (same place) Mumford sketches two more "coordinate oriented" methods
- Tracking the Weierstrass point of curves.
- Tracking invariants of the Chow form of the canonical curve (the Chow form is the equation for pairs of hyperplanes such that H_1 \cap H_1 \cap canonical-curve is not 0).
He also says these are the only "coordinate oriented" methods he knows of.