These were done by cutting shapes out of the finger painted paper, but I think it would work well to cut the shapes out of a heavier cardstock (either by hand, or using a diecut) and pasting that over the finger painting...like a mat.
How to keep a 1 year old busy -- cut a sheet of clear contact paper and tape it to the table STICKY SIDE UP. Give tot scraps of construction paper to stick to the contact paper. Great first craft and can be done independently with almost no mess!
I made "paint pops" for my toddler class. Frozen paint with a stick (clothes pins worked great) in the middle, made a great art activity for the little ones. Just make sure they know it is not food!! :)