Many 17th century settlers in what is now the United States were indeed indigent or criminals, but not all, and we should understand the "criminality" in question.
Many English farmers lost their livelihood due to enclosure, which had reached new heights during the Tudor years. Some ran themselves into debt and faced debtors' prison (indeed, Georgia Colony was originally intended to be a haven where debtors could work off their debts instead of wasting away in jail). At the same time, as the early colonies grew and prospered, they faced labor shortages. It is not surprising then that many companies were formed to ship surplus labor from the British Isles to North America, recouping the expense of transport by collecting the passenger's wages over seven years of indentured servitude.
Besides indebtedness, worshipping outside the current fashion of the Church of England could also land you in legal trouble. "Puritans" (encompassing a variety of Reformed theology groups), Baptists, Quakers, and Anabaptists were out of favor, not to mention Catholics, Lutherans, or Unitarians. Adherents of any minority religion would have seen a strong incentive to emigrate, and officials would have had an incentive to see them leave the country.
As for common criminals, to be sure, there would have been some. Certainly, people with such histories or inclinations would have seen an additional incentive to pursue a new life on another continent. On the other hand, the companies investing in colonies would have had little incentive to transport large numbers of unreliable petty thugs and thieves, the better to lose their investment. Neither were there large-scale deportations of criminals from England to the Americas: Pennsylvania was not Van Diemen's Land.
As always, we should take care not to over-generalize. The Pilgrim Fathers who settled Plymouth numbered barely 100, yet even they represented a cross-section of English commoners.
Some would have held comfortable places in English society were it not for their congregationalist religious beliefs. The first governor, John Carver, was a wealthy merchant. William Brewster, the religious elder, was a Cambridge-educated diplomat. Most of the other passengers were farmers or tradesmen, or of unknown occupation.
Not all who traveled on the Mayflower were religiously motivated, however. John Alden, by legend the first to step foot on Plymouth Rock, was a sailor in the employ of the Mayflower's owners. Myles Standish, while religiously sympathetic, was originally a hired gun for military and security affairs. Others were laborers recruited to work in the colony, or servants of the other settlers, and there were four children sent as indentured servants having been disowned as illegitimate.