Walnut handles on kitchen knives are almost worry-free anyway; not much reason to worry about them, as walnut is very stable. I'm sure that's at least partly why it's used so much on kitchen knives. My mother has used a set of Chicago Cutlery walnut-handled kitchen knives for the last 30-40 years. She used to use (I think) some vegetable oil and/or mineral oil (a.k.a. 'butcher block' oil) on the handles, though I don't think she's reapplied it in the last 5-10 years, maybe longer. In spite of that, they've held up fine. They do have some verdigris around the brass handle pins; I think that might've come from the acidic nature of the vegetable oil, as it degrades and reacts with the copper in the brass. Vegetable oil and other food-based oils also tend to get gummy/sticky when old; some of my mother's CC knives still have some stickiness in the handles in spite of having not been oiled in several years, at least.
If anything, mineral oil should do fine on them. The aforementioned Danish Oil should also do well with them; I used some of that on a Walnut-handled Opinel folder of mine, and it's worked out nicely. Once it cures in the wood, essentially nothing will erode it or remove it, short of sanding it out.
I'd also point out that washing the knives by hand, and NOT IN THE DISHWASHER is likely a good way to extend their useful life. Automatic dishwasher detergent is very caustic stuff, and can ruin most anything in the heat & humidity of the dishwasher.
David