Busy lizzies are an annual and therefore will die this winter.  I have heard of people taking cuttings of busy lizzies, but not tried it myself.

You would normally cut the stem between leaf nodules.  Then trim your cutting so that the end of the stem is just below a leaf nodule.  Remove the bottom leaves that will be in the water otherwise they go slimy.  The roots will grow from the leaf nodules.  Put them in a glass or jam jar of water on the kitchen windowsill and I think they say not to allow the end of your stalks to touch the bottom of the jar.  So have enough cuttings in there so that they are suspended in the neck of the jar and keep the water topped up of course.  When you come to pot them up you will need to very gently tease apart the roots, if like me you have left it too long before you pot them on!