John Morris: I recommend swimming around high tide (the river is tidal as far as Pulborough - four hours after Littlehampton).
High tide here is two hours after Littlehampton. Limited parking at the lane end, but more with a short walk from further up, enough for, say, the numbers a social swim might attract. For entry to the water, you can use the presently unused ferry crossing point, or, turning up-river or down, there other possible entry points, some of which may be used by fishers.
Above Bury however, the river becomes narrower and shallower and the banks are much muddier, especially as the tide goes down - but some may enjoy - mud between the toes is a price well worth paying, in my view.
There are footpaths along most of both sides of the river and several bridges over. I expect that most swimmers would get to the start points by car.
There are points to bear in mind for both. The Arun has the fastest tidal flow of any river in England. Swimming is much influenced by the state of the tide. There are thus several possible "styles" for a swim. 1. Go around the top of the tide for a leisurely swim from/around/back to the entry point. 2. Go, say, an hour before high tide and swim with the flow upstream and back as the tide turns. 3. Do a distance. Say, Bury to just north of Arundel (or vice-versa) and have a non-swimming partner meet you at the destination with your clothes
You are likely to be alone, housing distant, with only the occasional canoeist or boatie. (in summer, much of the bank is shielded by reeds - "skinny" is a possibility).