I do, but we have bamboo floors which stay cool, and of course hard. I just purchased some of those foam floor mats too. Great deal on Amazon, $15 for 24 sq ft. Hope they do their stuff?
I always did so if she spit-up on it I could just throw it in the wash and the carpet stayed clean. Now that she is crawling I don't. The blanket we used most was a fleece no-sew blanket so it was soft and the fringe edge was fun for her to play with.
Re: When DC plays on the floor, do you use a blanket?
ditto this