First, I'm no expert. But here is what has worked for me. I leave about an inch or so of the stem when I cut away the proliferation. This gives something to stick in the soil to help anchor the plantlet. There should be root nodules (?) at the base of the proliferation and you want these to have contact with the soil. I usually dip them in rooting hormone and stick them in regular potting soil. I'm very careful to keep the soil moist. In a few weeks the plant should form roots. I check by giving them a gentle tug. Results may be different for other folks, but for me it takes a couple years before the new plant blooms. Hope this helps and that you have good luck.

I have had NO luck propagated daylillies from cuttings or etc..... have had luck with division & seed

I used basically the same method as Opal, but I cut the stems a bit shorter and used gel4plug plant propagation gel instead of rooting hormone powder. Out of 12, 10 rooted and are doing well.
CLF

I share Opal's method. This year pickings are slim, guess it depends on the weather. I place mine in a styrofoam coffee cup and place them in a sunny window. The cutting may die back, but almost always there is new growth, and by early May a well established plant is ready to be placed in its permanent home. I usually get a flower stalk the year following placement in the permanent site. Having limited space, I give away most of my new items, which also include stock grown from seeds. In Fall 2013, I nipped a cutting from a plant at a local store which was set aside for disposition, unable to be sold. No flower stalk in 2014, but this year it had one stalk with lilac petals with a yellow center. On the stalk are two proliferates, so the process will hopefully be repeated, with one plant headed to Kentucky and the other locally here in Wisconsin. T.E. Hoffmann, Waukesha USA

I didn't know this was possible.

So I googled and found: http://msucares.com/news/print/sgnews/sg01/sg010827.html

Interesting stuff.