Two days ago I cut a plantlet off of the mother plant. The plantlet had one leaf and the mother has 2, a total of three leaves were on one plant before chopping. The newer leaf on the mother plant all of a sudden became thin and floppy the same day. Is this because of stress? Is the new leaf a goner?

I guess my question is....why did you feel the need to propagate an obviously small and immature plant in the first place? It should have been left to grow a lot larger and mature before you decided to propagate it.
When you say 'plantlet', what do you mean? If you mean 'offset', a one leaf offset should never be taken off of a 3 leaf plant.

I'm new to anthuriums. A friend who I thought was more experienced told me that you can cut the offset off once the leaf was thickened and had its own established root system. Which I did and the one I cut is the one that isn't thinning and floppy. The one to the right is the cut one (which is doing perfect). The floppy one is the one that was connected to the mother leaf that is about a year old on the left. I was just following advice of someone who knew more than I did.

Hmmm, well, its a case of just because you can does not always mean you SHOULD. Allowing your plants to mature before propagating them is really the best thing you can do. Crystallinum eventually get leaves that can be 15-18" long, and when the offsets are larger they will do better than very small offsets. It may just be sulking. Some plants do not like to be disturbed that way. My Anthurium papillilaminums hate to be propagated and can sulk for a month sometimes.

The proper way to propagate an anthurium offset is to unpot the entire plant and separate the two plants by separating the root system manually, not by slicing it off. Both the root system of the mother and the onset are then preserved. You may have damaged the root system of the 'floppy' one and it may be showing its unhappiness

I'm new to anthuriums. A friend who I thought was more experienced told me that you can cut the offset off once the leaf was thickened and had its own established root system. Which I did and the one I cut is the one that isn't thinning and floppy. The one to the right is the cut one (which is doing perfect). The floppy one is the one that was connected to the mother leaf that is about a year old on the left. I was just following advice of someone who knew more than I did.