(source: https://jsfiddle.net/b5fr05p4/)

fiddle.js

class Form extends React.Component {

  constructor (props) {
     super(props)
     this.state = {
       input: ''
     }
  }

handleChangeInput = (text) => {
    this.setState({ input: text })
  }
  
  render () {
    const { input } = this.state

    return (
       <View>
          <TextInput style={{height: 40, borderColor: 'gray', borderWidth: 1}}
            onChangeText={this.handleChangeInput}
            value={input}
          />
        </View>
      )
    }
 }