In "A Worn Path" how do the themes stand the test of time and relate to today?
I think one way of answering this question is to look at the story as not something that is based in one contextual, historical period alone, but as offering us an archetype. The archetype of a journey is one that occurs many, many times in a variety of forms, as one central character tries to complete a journey, but encounters many barriers and obstacles on the way. You only have to consider characters such as Odysseus in the Odyssey to think of an excellent example. In the same way, Phoenix Jackson is a character with a definite goal, but she has to encounter and overcome a variety of obstacles on the path to her goal.
If we consider this archetypal reading, we can see that the themes presented in this excellent short story are truly timeless. All of us, at one stage or another, have to engage on some kind of journey or a quest: it may be to complete our studies or to get to a particular position in a job, but the archetype of a journey is one that is truly timeless and that we can all connect to. Often, as in the case of Phoenix Jackson, we set out on our own "worn path" for love, as she does for her grandson. The love that we have for another person often gives us the strength and endurance that we need to overcome the barriers on our "worn path," as love gives us strength.
The obstacles that plague Phoenix Jackson on her journey are emblematic of the kinds of obstacles that have always, and will always, exist in the human experience.
Phoenix faces racism, ageism, classism, and sexism on her quest to secure medicine for her grandson. These forces of discrimination touch all lives. At one time or another in our life's journey, we are mistreated, looked down on, or dismissed by other people and social forces. It is the human condition. And in many cases, we have to carry burdens for other people in our lives besides ourselves. This is certainly true with regard to what Phoenix is trying to accomplish.
The flip side of these truths that Welty observes is that it is up to us to push through the forces that work against us. Phoenix endures what she must and retains her self-worth and dignity; she is wise in her approach and does not push back against the things that she cannot change. She does what so many of us do when we meet an obstacle: she finds a workaround.