Passing drawer navigator to the stack navigator header is pretty easy, though it might seem difficult. It is seen by the number of questions asked on programming portals. Therefore, I will present here a shortened version of the code that will show how to achieve this effect.

In drawer nawigator we set stack nawigator as main screen (Home)

DrawerNavigator.js

const MyDrawerNavigator = createDrawerNavigator({
    Home: {screen: StackNavigator},
    ...
},
{
    initialRouteName: 'Home',
    contentComponent: ContentComponent,
    drawerWidth: Dimensions.get('window').width,
    drawerPosition: 'left',
    drawerBackgroundColor: 'transparent',
}
);

const AppContainer = createAppContainer(MyDrawerNavigator);

export default class DrawerNavigator extends Component {
    render() {
        return <AppContainer />;
    }
}

Do stack nawigatora przekazujemy opcję otwarcia drawer navigatora w postaci screenProps.

In screenProps we pass open drawer nawigator function to stack navigator

StackNavigator.js

const AppContainer = createAppContainer(MyStackNavigator);

export default class StackNavigator extends Component {
  render() {
    return <AppContainer screenProps={{ openDraw: () => this.props.navigation.dispatch(DrawerActions.openDrawer()) }} />;
  }
}

Then, in the main screen of the navigator, we read screenProps and assign open drawer navigator option to created button .

HomeScreen.js

static navigationOptions = ({ screenProps }) => ({
      title: 'Home',
      headerLeft: (
          <Button onPress={() => screenProps.openDraw() }
              title="OPEN"
              color="red"
          />
      ),
})

Efect: