While Cromwell certainly supported the move, and subsequent laws imposing penalties for those who continued to enjoy Christmas, he does not seem to have played much of a role in leading the campaign.
Throughout the medieval period, Christmas Day had been marked by special church services, and by magnificent feasts accompanied by heavy drinking. The subsequent 12 Days of Christmas saw more special services along with sports, games and more eating and drinking.
By the early 17th Century Puritans and other firm Protestants were seeing the Christmas jollifications as unwelcome survivors of Catholicism as well as excuses for all manner of sins.
There was a widespread, though minority view, that Christmas should be a fast day devoted to sober religious contemplation. The defeat of King Charles I in the Civil War put the more extreme Protestants into power and so Parliament passed a series of measures to enforce this campaign on others.
The legislation was deeply unpopular and was enforced only sporadically. When King Charles II returned to power in 1660 one of his first acts was to repeal all the anti-Christmas legislation, helping foster his image as the “Merry Monarch”.