privatereposbasic.java

//First we build the URI and the auth header and pass them to the AsyncTask

private static final String GIT_HUB_API = "api.github.com";
private static final String USER_ENDPOINT = "user";
private static final String REPOS_ENDPOINT = "repos";
private static final String ERROR = "error";

//(...)

Uri.Builder builder = new Uri.Builder();
builder.scheme("https")
    .authority(GIT_HUB_API)
    .appendPath(USER_ENDPOINT)
    .appendPath(REPOS_ENDPOINT);
    
//this is how we build the Basic Authentication header contents
String base = Base64.encodeToString((user + ":" + pass).getBytes(), Base64.NO_WRAP);
String authHeader = "Basic " + base;    
    
new NetworkWork().execute(builder.build().toString(), authHeader);

//(...)

//Then we build the actual AsyncTask
class NetworkWork extends AsyncTask<String, Void, String> {
  @Override
  protected String doInBackground(String... url) {
    if (url == null || url[0] == null) return ERROR;

    OkHttpClient client = new OkHttpClient();

    Request.Builder builder = new Request.Builder().url(url[0]);
    //here we add the Basic Authentication header
    if (url.length > 1) builder.addHeader("Authorization", url[1]);

    try (Response response = client.newCall(builder.build()).execute()) {
        if(!response.isSuccessful()) return ERROR
      
        return response.body().string();
    } catch (IOException e) {
        e.printStackTrace();
        return ERROR;
    }
  }

  @Override
  protected void onPostExecute(String s) {
    if (s.equals(ERROR)) {
        Toast.makeText(MainActivity.this, "Cannot fetch data from GitHub!", Toast.LENGTH_SHORT).show();
    } else decodeJson(s);
  }           
}

//Finally we decode the JSON (see above)