how do I style an Alert element in react-native?

You can use a custom library like react-native-modalbox.

You'll be able to style the modal as you like:

        <Modal style={{background: 'red'}} ref={"modal1"}>
          <Text style={{color: 'white'}}>Basic modal</Text>
          <Button onPress={this.toggleSwipeToClose} style={styles.btn}>Disable swipeToClose({this.state.swipeToClose ? "true" : "false"})</Button>
        </Modal>

You can open the modal using

openModal1(id) {
    this.refs.modal1.open();
  }

Check out the example to see more options.

Bonus: If you want to find a good library for react-native, try js.coach

There actually is a way to customize the text on the buttons but you're limited to what is provided... Below is the type definition from react native. Then a simple example showing a red button. Basically the "default" is blue, "cancel" is bold but still blue and "destructive" is red.

**
 * An Alert button style
*/
export type AlertButtonStyle = $Enum<{
/**
* Default button style
*/
'default': string,
/**
* Cancel button style
*/
'cancel': string,
/**
* Destructive button style
*/
'destructive': string,
}>;


Alert.alert("Look out!",
        "Hitting reset will wipe all the app's data on your phone. This cannot be undone!",
        [
        {text: 'Reset', onPress: this._doSomethingSerious, style: 'destructive'},
        {text: 'Cancel'},
        ],
        {cancelable: false}
    )