Same engine as Half-Life, same rules: open engine, files you must own.
Counter-Strike 1.6 runs on GoldSrc, which means the same Xash3D browser builds that run Half-Life can run CS 1.6, provided you supply the game's own files.
Counter-Strike is now free on Steam, which changes the calculation usefully. The cheapest legal path is to install it, and then the browser builds have something legitimate to load from.
Servers are still running. The community never let this one die.