The puritans were in a constant battle with what, was it with crime, corruption, the government, or impurities?
The Puritans were not in a constant struggle with Satan, any more than other societies were. The only real "battle" with the devil and his minions was during the brief period marked by the Salem Witch scare, which was geographically limited and (compared to witch scares in Europe) short-lived.
The Puritans did not "struggle" against anything that was not a matter of struggle for others attempting to build a new society in a primitive environment. They did, however, aspire to set themselves up as a model to the world of a society defined by its Christian charity. This goal was instituted by John Winthrop in a sermon delivered on board the Arbella before they ever landed in America:
we must be knit together, in this work, as one man. We must entertain each other in brotherly affection. We must be willing to abridge ourselves of our superfluities, for the supply of others’ necessities. We must uphold a familiar commerce together in all meekness, gentleness, patience and liberality. We must delight in each other; make others’ conditions our own; rejoice together, mourn together, labor and suffer together, always having before our eyes our commission and community in the work, as members of the same body. So shall we keep the unity of the spirit in the bond of peace. The Lord will be our God, and delight to dwell among us, as His own people, and will command a blessing upon us in all our ways, so that we shall see much more of His wisdom, power, goodness and truth, than formerly we have been acquainted with. We shall find that the God of Israel is among us, when ten of us shall be able to resist a thousand of our enemies; when He shall make us a praise and glory that men shall say of succeeding plantations, "may the Lord make it like that of New England." For we must consider that we shall be as a city upon a hill. The eyes of all people are upon us. So that if we shall deal falsely with our God in this work we have undertaken, and so cause Him to withdraw His present help from us, we shall be made a story and a by-word through the world. We shall open the mouths of enemies to speak evil of the ways of God, and all professors for God's sake. We shall shame the faces of many of God's worthy servants, and cause their prayers to be turned into curses upon us till we be consumed out of the good land whither we are going.
Contrary to popular belief, Puritans were not prudish; they wore colorful clothes, regularly consumed alcohol, and did not eschew fun and good times. Rather they believed one must practice moderation in all things, including food and drink. The only passion to which they were to "zealously aspire" was the practice of Christian piety. Their "struggles" were no more than one might find in any other settlement; in fact they perhaps struggled less than Jamestown. There were no rebellions in New England; Jamestown was rocked by a rebellion led by Nathaniel Bacon.
The Puritans were in a constant struggle for survival against the ultimate evil, Satan!
Their belief was that they were God's instruments in the New World, and were traveling there to help create a kingdom of heaven on earth. When the first party of puritan settlers, under the leadership of John Winthrop, was about to leave for Massachusetts Bay, they were addressed by the Reverned John Cotton, who said that they were God's chosen people like the jews of the Old Testament.
This is important to note because since they were God's people, anything that happened to them that was bad was interpreted as the work of the devil. They saw themselves on the front lines of his battle for the New World, and they were the right hand of God. So when the Native Americans rebelled because of sickness or land theft, they were seen as the agents of Satan, and were targeted for summery distruction. When droughts hit, it was the devil trying to starve them out. When crimes occured, or when government offcials were caught taking bribes, they were agents of Satan working is disguise, and needed to be punished in a fitting manner.
This was also the reason why they were so quick to believe in witches and sorcery.