Based on my preliminary Internet research, you can't propagate raspberries from cuttings the common way, i.e. cutting pieces from the stem and placing it in water or a rooting medium. I tried it myself with no success (I have near 100% success rate on propagating citrus, blueberries, and many other plants ... just so you know that I have some experience in the matter). I wonder if anyone had any success on this, and if so, how (rooting medium, rooting hormone, humidity, temperature, lighting, ....)?
You might be able get a cutting to root in medium, but frankly that's doing things the hard way with raspberries.
If you are trying to propagate red raspberries, you would probably be better off dividing a sucker shoot off of the main plant at the roots. With a shovel or a spade you can split a sucker off of the main root mass (be sure to get some of the root mass) and have a new plant to place as you wish. Reds produce suckers in the fall, best time to divide off is early spring.
For black raspberries, you can use tip layering. Some varieties of black raspberries are so easy to propagate this way that they do it by themselves. For tip layering you bend a branch down so that the tip of the stem touches the soil, then pile a bit of soil over top to hold it there. The tip will form roots at the point of contact, next spring the new plant can be cut from it's older stem and now you have a new separate plant.