Use @trincot's solution if you really don't care about which node/value matched. It's straightforward, well-written, and solves your problem very effectively.
If you want more than just a Boolean value as the result of your digging, read along ...
I really doubt your need for this, but if your object is significantly large, you will want an early exit behaviour – what this means is that as soon as a match is found, iteration through your input data will stop and true/false result will be returned immediately. @trincot's solution offers early exit, but solutions that using map, filter, or reduce offer no such behaviour.
findDeep is much more useful than just checking if a string value starts with another string value – it takes a higher-order function that is applied for each leaf node in your data.
This answer uses my findDeep procedure to define a generic anyStartsWith procedure by checking if findDeep returns undefined (no match)
It will work any any input type and it will traverse Object and Array child nodes.
const isObject = x=> Object(x) === x
const isArray = Array.isArray
const keys = Object.keys
const rest = ([x,...xs]) => xs
const findDeep = f => x => {
  
  let make = (x,ks)=> ({node: x, keys: ks || keys(x)})
  
  let processNode = (parents, path, {node, keys:[k,...ks]})=> {
    if (k === undefined)
      return loop(parents, rest(path))
    else if (isArray(node[k]) || isObject(node[k]))
      return loop([make(node[k]), make(node, ks), ...parents], [k, ...path])
    else if (f(node[k], k))
      return {parents, path: [k,...path], node}
    else
      return loop([{node, keys: ks}, ...parents], path)
  }
  
  let loop = ([node,...parents], path) => {
    if (node === undefined)
      return undefined
    else
      return processNode(parents, path, node)
  }
  
  return loop([make(x)], [])
}
const startsWith = x => y => y.indexOf(x) === 0
const anyStartsWith = x => xs => findDeep (startsWith(x)) (xs) !== undefined
let obj = { 
  'child' : {
    'child_key': 'asdfghhj'
  },
  'free': 'notasd',
  'with': 'asdhaheg'
}
console.log(anyStartsWith ('asd') (obj))   // true
console.log(anyStartsWith ('candy') (obj)) // false
 
 
You'll see this is kind of a waste of findDeep's potential, but if you don't need it's power then it's not for you.
Here's the real power of findDeep
findDeep (startsWith('asd')) (obj)
// =>     
{
  parents: [
    {
      node: {
        child: {
          child_key: 'asdfghhj'
        },
        free: 'notasd',
        with: 'asdhaheg'
      },
      keys: [ 'free', 'with' ]
    }
  ],
  path: [ 'child_key', 'child' ],
  node: {
    child_key: 'asdfghhj'
  }
} 
The resulting object has 3 properties
- parents– the full object reference to each node in the matched value's lineage
- path– the path of keys to get to the matched value (stack reversed)
- node– the key/value pair that matched
You can see that if we take the parent object as p and reverse the path stack, we get to the matched value
p['child']['child_key']; //=> 'asdfghhj'