Besides the Jedi force of around 10,000, there was the Republic's non-Jedi law enforcers, the Judicials.
A Judicial was a member of the Judicial Department. They served as the primary non-Jedi peace officers in the Galactic Republic. Most Judicials attended the Judicial Academy before being assigned to the Judical Forces. With the start of the Clone Wars, the Republic formed the Grand Army of the Republic and the Republic Navy which took over the military functions performed by the Judicial Department. Many admirals, generals, and other commanders were drawn from the ranks of the Judicials. - Judicial, Wikia
The prime example is Tarkin (from which the above quote lists as a citation). He served on the Judicials before the war, then was later made a Captain in the Republic Navy. - Star Wars: Tarkin.
Systems also had their own navies and security forces. You see this with worlds like Mandalore in Clone Wars.
Most of the graduates [of Sullust Sector Spacefarers Academy] would be moving on to positions in commercial piloting, local system navies, or the Judicial Department. - Tarkin, ch. 6