Depends on where you were. I can't speak for the southern colonies (which may seem odd, since that's closer to my area), but in the Puritan colonies, the English Civil War was hailed with great enthusiasm. At this time the Puritans were not separatists, but reformers. Their self-stated goal was to create a godly society and then export it back to England, to the benefit of all. When Cromwell's Civil War kicked off, they thought that they were witnessing the beginning of a worldwide Puritan society. The Anglican church had treated Puritans pretty roughly so many of them were thrilled to see Cromwell gain power, especially since he was a Puritan too. Some Puritans even left New England and traveled back to England to participate in the Civil War.

Unfortunately for them, the crown came back into power some ten years later. This made the Puritans very despondent and they began to seriously consider separation rather than reform. This was exacerbated when the crown revoked their charter in 1684. There was even a minor rebellion where they basically kicked out the crown appointed governor.

All in all, the Puritans were pleased with the English Civil War and did their best to support it. Unfortunately, I'm not personally familiar with a work that specifically looks at this topic.

Sources: Anderson, New England's Generations, Hall, A Reforming People, Innes, Creating the Commonwealth.

since onthefailboat game a NE answer let me wade out into the impact on Virginia the flagship state university's nickname is the Cavaliers so named because of the large number of royalist supporters emigrees during the civil wars (it was also the only royally administered colony at the time) including declaring for Charles II in 49 and creating a historic self image of Virginia elites as Cavaliers (though this self image neglects that before the execution of Charles I VA declared itself neutral and most emigres of this period were indentured servants not royalty).

Here's a nice high quality wiki page from the Virginia Foundation for the Humanities

http://www.encyclopediavirginia.org/English_Civil_Wars_and_Virginia_The

the timeline especially does a nice job of laying out the impact of the civil war on VA events (including a skirmish on the James river)

also further reading list there points to this oldish book which seems to answer the main question

Bliss, Robert M. Revolution and Empire: English Politics and the American Colonies in the Seventeenth Century. New York: Manchester University Press, 1993.