The most practical solution is to use a library for this like react-measure.
Update: there is now a custom hook for resize detection (which I have not tried personally): react-resize-aware.  Being a custom hook, it looks more convenient to use than react-measure.
import * as React from 'react'
import Measure from 'react-measure'
const MeasuredComp = () => (
  <Measure bounds>
    {({ measureRef, contentRect: { bounds: { width }} }) => (
      <div ref={measureRef}>My width is {width}</div>
    )}
  </Measure>
)
To communicate size changes between components, you can pass an onResize callback and store the values it receives somewhere (the standard way of sharing state these days is to use Redux):
import * as React from 'react'
import Measure from 'react-measure'
import { useSelector, useDispatch } from 'react-redux'
import { setMyCompWidth } from './actions' // some action that stores width in somewhere in redux state
export default function MyComp(props) {
  const width = useSelector(state => state.myCompWidth) 
  const dispatch = useDispatch()
  const handleResize = React.useCallback(
    (({ contentRect })) => dispatch(setMyCompWidth(contentRect.bounds.width)),
    [dispatch]
  )
  return (
    <Measure bounds onResize={handleResize}>
      {({ measureRef }) => (
        <div ref={measureRef}>MyComp width is {width}</div>
      )}
    </Measure>
  )
}
How to roll your own if you really prefer to:
Create a wrapper component that handles getting values from the DOM and listening to window resize events (or component resize detection as used by react-measure).  You tell it which props to get from the DOM and provide a render function taking those props as a child.
What you render has to get mounted before the DOM props can be read; when those props aren't available during the initial render, you might want to use style={{visibility: 'hidden'}} so that the user can't see it before it gets a JS-computed layout.
// @flow
import React, {Component} from 'react';
import shallowEqual from 'shallowequal';
import throttle from 'lodash.throttle';
type DefaultProps = {
  component: ReactClass<any>,
};
type Props = {
  domProps?: Array<string>,
  computedStyleProps?: Array<string>,
  children: (state: State) => ?React.Element<any>,
  component: ReactClass<any>,
};
type State = {
  remeasure: () => void,
  computedStyle?: Object,
  [domProp: string]: any,
};
export default class Responsive extends Component<DefaultProps,Props,State> {
  static defaultProps = {
    component: 'div',
  };
  remeasure: () => void = throttle(() => {
    const {root} = this;
    if (!root) return;
    const {domProps, computedStyleProps} = this.props;
    const nextState: $Shape<State> = {};
    if (domProps) domProps.forEach(prop => nextState[prop] = root[prop]);
    if (computedStyleProps) {
      nextState.computedStyle = {};
      const computedStyle = getComputedStyle(root);
      computedStyleProps.forEach(prop => 
        nextState.computedStyle[prop] = computedStyle[prop]
      );
    }
    this.setState(nextState);
  }, 500);
  // put remeasure in state just so that it gets passed to child 
  // function along with computedStyle and domProps
  state: State = {remeasure: this.remeasure};
  root: ?Object;
  componentDidMount() {
    this.remeasure();
    this.remeasure.flush();
    window.addEventListener('resize', this.remeasure);
  }
  componentWillReceiveProps(nextProps: Props) {
    if (!shallowEqual(this.props.domProps, nextProps.domProps) || 
        !shallowEqual(this.props.computedStyleProps, nextProps.computedStyleProps)) {
      this.remeasure();
    }
  }
  componentWillUnmount() {
    this.remeasure.cancel();
    window.removeEventListener('resize', this.remeasure);
  }
  render(): ?React.Element<any> {
    const {props: {children, component: Comp}, state} = this;
    return <Comp ref={c => this.root = c} children={children(state)}/>;
  }
}
With this, responding to width changes is very simple:
function renderColumns(numColumns: number): React.Element<any> {
  ...
}
const responsiveView = (
  <Responsive domProps={['offsetWidth']}>
    {({offsetWidth}: {offsetWidth: number}): ?React.Element<any> => {
      if (!offsetWidth) return null;
      const numColumns = Math.max(1, Math.floor(offsetWidth / 200));
      return renderColumns(numColumns);
    }}
  </Responsive>
);