The Mishna permits (according to R. Meir) a person who has an amputated leg to go out with his "kav" - prosthetic leg. Tosafos has different approaches as to what function the kav actually served. Tosafos assumes that if this is the his primary method of moving around, we can prove from here that an amputee can also use a crutch or crutches which is his primary method of movement. However, if the prosthetic leg that the mishna is speaking about is more for appearance and aesthetics, we can't draw a proof from here regarding the use of a cane or crutches.
The Shulchan Aruch 301:17 paskens that a person who can walk without crutches may go out with crutches on shabbos. The assumption is that since he can't move without them, they are considered a "shoe" for him and permitted to be used on Shabbos. But the Shulchan Aruch continues that if he can move without them and is only using them to strengthen himself or hold himself better, it is forbidden. It isn't clear from the Shulchan Aruch whether the problem is that since he can move without it, we are afraid he may come to carry it, or since he can move without it, it's not considered a shoe hence a Torah violation of carrying. The Biur Halacha cites the Levush who says that on a Torah level it is not considered carrying since he is leaning on it, but on a Rabbinic level we are concerned that since he doesn't need it that desperately, he will come to carry it. However, the Biur Halacha rejects this approach based on the gemara 66b. The gemara says that a cane used by the elderly is to enable him to walk straight and straighten himself up since his legs shake, but it isn't made for the leaning of one's entire body and therefore it is not tamei as a midras. From here we see that something which is used to strengthen oneself and not to fully lean on, such as a cane (vs. a crutch), is not tamei midras and therefore on shabbos should be considered carrying. Therefore, the M.B. (64) paskens that an elderly person who can walk around inside his home without a cane, but when he goes out he uses a cane for the security and sturdiness, it is forbidden and is considered a משוי. The M.B. holds that it's not only a concern that he may come to carry it, but is actually an issur d'oraysa of carrying. However, if he is so old that he can't even move around inside without it, it becomes like the kav of a amputee that can be used in the reshus ha'rabim.
Based on this, those people who use walkers to enable themselves to walk, the halacha would be that if they require those walkers inside, they can use it outside as well. But if they only use it when going out to secure themselves and prevent themselves from falling, it is assur to go out with it on shabbos when there is no eiruv.