This is a workaround for the buggy react-native TextInput multiline on Android. Inspired by the comments on https://github.com/facebook/react-native/issues/12717. Based the work of @catchin with the addition of handling the state via the container.

MultilineTextInput.js

import React, { PropTypes, PureComponent } from 'react';
import { TextInput } from 'react-native';
import debounce from 'debounce';

/**
 * This is a workaround for the buggy react-native TextInput multiline on Android.
 *
 * Can be removed once https://github.com/facebook/react-native/issues/12717
 * is fixed.
 *
 * Example for usage:
 *   <MultilineTextInput value={this.state.text} onChangeText={text => setState({text})} />
 */
export default class MultilineTextInput extends PureComponent {
    constructor(props) {
        super(props);
        this.state = { selection: { start: 0, end: 0 } };
        // Prevent 2 newlines for some Android versions, because they dispatch onSubmitEditing twice
        this.onSubmitEditing = debounce(this.onSubmitEditing.bind(this), 100, true);
    }
    componentWillReceiveProps(props) {
        if (props.selection) {
            this.setState({ selection: props.selection });
        }
    }

    focus() {
        this.refs.text.focus();
    }

    onSubmitEditing() {
        const { selection } = this.state;
        const { value } = this.props;
        const newText = `${value.slice(0, selection.start)}\n${value.slice(selection.end)}`;
        
        this.props.onChangeText(newText);
        
        // move cursor only for this case, because in other cases a change of the selection is not allowed by Android
        if (selection.start !== this.props.value.length && selection.start === selection.end) {
            const newSelection = {
                selection: {
                    start: selection.start + 1,
                    end: selection.end + 1,
                },
            };
            if (this.props.onSelectionChange) {
                this.props.onSelectionChange({ nativeEvent: newSelection });
            } else {
                this.setState(newSelection);
            }
        }
    }
    render() {
        return (
            <TextInput
                ref="text"
                multiline
                blurOnSubmit={false}
                selection={this.state.selection}
                value={this.props.value}
                onSelectionChange={event => this.setState({ selection: event.nativeEvent.selection })}
                onChangeText={this.props.onChangeText}
                onSubmitEditing={this.onSubmitEditing}
                {...this.props}
            />
        );
    }
}

MultilineTextInput.propTypes = {
    value: PropTypes.string.isRequired,
    onChangeText: PropTypes.func.isRequired,
};

Thank you very much for this workaround. :)
It was a great help and we used it in our project work.