MyContainerComponent.jsx

import React from 'react'
import { shallow } from 'enzyme'

const flushPromises = () => new Promise(resolve => setImmediate(resolve))

const fetchTheData = jest.fn()

class MyContainerComponent extends React.Component {
  state = { error: null }

  async componentDidMount() {
    try {
      await fetchTheData()
    } catch (e) {
      this.setState({ error: 'There was a problem' })
      throw e // rethrow so it gets handled by a generic unhandled promise rejection handler (e.g. Sentry)
    }
  }

  render() {
    return <p>{this.state.error}</p>
  }
}

it('should handle errors fetching the data', async () => {
  fetchTheData.mockImplementation(() => Promise.reject(new Error('something bad happened')))

  const component = shallow(<MyContainerComponent />)

  await flushPromises()
  expect(component.update()).toIncludeText('There was a problem')
})