The classical (pre Deligne-Mumford) approach is to map $\mathcal{M}_g$ into $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}_g$ is quasi-projective is to define it as the Siegel upper half space $\mathcal{H}_g$ ($g$ by $g$ complex matrices with positive define imaginary part), modulu $\mathrm{SP}(2g,\mathbb{Z})$; We define the level cover $\mathcal{A}_g(m)\to\mathcal{A}_g$, which is the moduli of $\mathcal{A}_g$ plus torsion points, which is the quotient of $\mathcal{H}_g$ by $\Gamma(m)$ (the matrixes in $\mathrm{SP}(2g,\mathbb{Z})$ which are trivial modulo $n$), and send $\mathcal{A}_g(m)$ to some projective space using polynomials in the theta constants.
Reference (for both Torreli and the embedding of $\mathcal{A}_g$ above) : 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.