They were alarmed about secularism, though they would have called it infidelity . In their day, the great secularism was a form of deism that denied the divinity of Christ and undermined intimacy with God. To understand the Puritans, you have to adopt their attitude that life is a great adventure. The Puritans also feared the rising generation would not measure up to the piety of their fathers and mothers. They often talked about loss of faith in their children.
Life is a great adventure.
The Puritans saw themselves on a group mission, like a corporate Pilgrim's Progress. If you read the Puritans' writings as cold, theological prose, they will kill you quickly. Creating this was the adventure of a lifetime. There was persecution in England, but it was limited mostly to ministers. So it wasn't fear of persecution that drove the laity to come.