I have been using a wood hardener and so far that has been working. Should I be using a stabilizer instead?
More generally put: when fabricating a walking stick or walking cane what is the best product to use to stop the ends from splitting? A stabilizer or a hardener?
Generally the bottom end of a cane or stick is protected by a ferrule1 or full cap, traditionally of metal (hence the hard tap, tap, tap of a Victorian or Georgian gent walking down the street in movies). These days caps are often of rubber and a Google search for "walking stick caps" will bring up many vendors. There are numerous styles available in multiple diameters to suit sticks of all sizes and types.
As for the top end, many sticks feature caps or handles made from other materials which are glued in place (usually with a dowel or metal rod for reinforcement) and therefore help stabilise that area. If the top is just the cut end of the stick though any decent finish will protect it just fine. However, even if left bare there shouldn't be a particular issue with splitting if the wood is well dried beforehand2.
If your sticks feature simple cut tops you can continue to treat the end grain there with wood hardener for added insurance against splitting, nothing wrong with erring on the side of caution, but it shouldn't normally be necessary.
There's a good, if brief, overview of traditional stickmaking on the Woodworker's Institute site, Learn the Basic Rules of Stickmaking.
1 This can be as simple as a short cutoff of copper pipe.
2 One of the hardest parts of traditional stickmaking is having the patience to let your collected wood season, it takes months at minimum and possibly a year or more depending on species and the thickness of the branch or sapling.
If it is a walking stick* as apposed to a cane, one of the traditional ways of hardening the ends is with fire^. CA glue will also work to harden and stabilize the wood. As Graphus said a ferrule is excellent solution and can be decorative.
*A walking stick usually maintains the natural shape of the stick and in my view have the bark removed and may have branch nubs smoothed out and some what primitive looking.
^Heating the ends after initial shaping where some limited charing occurs witch can mostly be removed by sanding or carving.
