const myObj = {
  id: 1,
  anyProp: [{
    id: 2,
    thing: { a: 1, id: 10 },
    children: [{ id: 3 }]
  }, {
    id: 4,
    children: [{
      id: 5,
      children: [{
        id: 6,
        children: [{ id: 7 }]
      }]
    }]
  }]
};

const getValues = prop => obj => {
  if (!Object.keys(obj).length) { return []; }

  return Object.entries(obj).reduce((acc, [key, val]) => {
    if (key === prop) {
      acc.push(val);
    } else {
      acc.push(Array.isArray(val) ? val.map(getIds).flat() : getIds(val));
    }
    return acc.flat();
  }, []);
}

const getIds = getValues('id');

console.log(getIds(myObj));