Some information is in Dixon--Mortimer (1996), Chapter 8.5, "Maximal subgroups of the symmetric groups". Unfortunately:
The situation for infinite symmetric groups is more complicated, and it seems unlikely that there is a satisfactory description of the maximal subgroups in this case.
I do not think the situation has changed dramatically since 1996.
These lecture notes of Peter Neumann are also worth looking at: https://arxiv.org/abs/2307.11564