I don't know of any modification really making the game "high-res". Unmodified the game runs at a resolution of 640x480 pixels with 256 colors. As far as I know these values aren't part of any configuration file you could change. Instead, these are hardcoded, i.e. they're written into the code. To modify them, you'd have to change the right parts of the code, which would cause more issues: Other parts of the code might not get adjusted accordingly (e.g. offsets for the game UI, clipping code, etc.), which could cause image artifacts, misaligned UI elements, etc. - in short: a game you can't enjoy.
There are some "hacks" out there claiming to make the game high-res, but I guess all or at least most of them are either fake to steal other game accounts or they're just using a trick to fake higher resolutions: They take a screenshot, move the camera, take another screenshot, etc. and then use these screenshots to stitch them together creating a higher resolution image that is then presented to the player.
Things you can try to make the game screen appear smoother (I assume you don't like your current upscaling and don't want more viewspace):
- Change your driver/hardware settings to use different methods to upscale the screen image. These options can be hidden in your monitor's settings or in your graphic card settings (e.g. for Nvidia cards). E.g. you might be able to enable or disable code trying to blur edges on upscaled pixels.
- Disable scaling for your screen and graphics card (similar to previous point): This way you should get a crystal sharp game image (although it might be a bit small).
- Play in windowed mode (while I don't think there's any official way to do it, there are some tools allowing you to do so). Keep in mind that this is considered a "hack" and might result in your account and/or CD key(s) being banned.
As someone in the comments mentioned a mod for Diablo 2 allowing higher resolutions: You can't compare the two games. Starcraft is a classic 2D game only drawing a specific width/height on screen. Diablo 2 on the other hand is a 3D game (even though you're able to disable perspective etc.), so it probably places things a lot different (calculating offsets, limits, etc.).