Very close, indeed. In fact, both were heavily influenced by Middle French, the main difference between German and English. They were just influenced in different ways.
English is a strange concoction of Anglo-Saxon, a pinch of Welsh, plus a little Old Norse in the Viking age (giving us much of our word order), followed by Norman French for a few hundred years along with liturgical Latin for a semi-romantic flare, and some Greek, Arabic, Hebrew, Russian, Irish, Turkish, and many others, for good measure, to give us a robust vocabulary and too many ways to say the same thing. This goes far to explain why we have the longest dictionaries of any language.
And yet, even after all these influences, German and Dutch are certainly the closest to English in terms of shared vocabulary out of all languages in the world. But all Germanic or Romantic languages have much in common, and every Indo-European language has a little in common.
If you want to see more clearly how similar English and Dutch are, try reading a single sentence in Turkish (an Altaic language) or even Welsh (an Indo-European language spoken right next to England) and see how much you understand. Dutch is comparatively much easier to figure out.
Turkish: Merhaba, senin adın ne? (Hallo, hoe heet je?)
Welsh: Yr wyf flynyddoedd pedwar ar hugain oed (Ik ben vierentwintig jaar oud)
Doesn't Dutch seem easy now? :P