How can I get the Typescript compiler to output the compiled js to a different directory?

Use the option --outDir on tsc (configured within the File Watcher in IntelliJ)

From the command line documentation

--outDir DIRECTORY Redirect output structure to the directory.

Edit

Since Typescript 1.5, this can also be set in the tsconfig.json file:

"compilerOptions": {
    "outDir": "DIRECTORY"
    ...

Or, add "outDir": "build"to tsconfig.json file

Though these answers are correct you should consider whether you actually just want to hide your .js files from your IDE.

In Visual Studio Code, go to File > Preferences > Settings or your .vscode\settings.json file and enter:

"files.exclude": {
    "**/.git": true,
    "**/.DS_Store": true,
    "**/*.js" : {
        "when": "$(basename).ts"
    },
    "**/*.js.map": {
        "when": "$(basename)"
    }
}

The above hides .js files where a corresponding .ts file exists.