Within the Austen family there was a preference for using spas for ill health and visiting the seaside for pleasure. Edward Austen visited and James Leigh-Perrott lived in Bath for treatment of their gout. Jane and Cassandra Austen visited Cheltenham in 1816 to try to cure Jane's declining health. Their visits to the seaside were planned as recreational visits only, with no specific medical purpose attached to them. It was only the prospect of annual visits to the seaside that made the move to Bath tolerable to Jane.
In fact, during the Regency Era few men and women wore bathing costumes. They often swam nude, and entered the sea in separate beaches sheltered by rented bathing machines drawn by horses (much like those in the photograph above, taken in 1885-1890). Bathers changed in and out of their clothes in these portable dressing rooms. An 18th century description of a bathing machine from The Expedition of Humphrey Clinker gives one a good idea of what being in one was like:
Imagine to yourself a small, snug, wooden chamber, fixed upon a wheel-carriage, having a door at each end, and on each side a little window above, a bench below - The bather, ascending into this apartment by wooden steps, shuts himself in, and begins to undress, while the attendant yokes a horse to the end next the sea, and draws the carriage forwards, till the surface of the water is on a level with the floor of the dressing-room, then he moves and fixes the horse to the other end - The person within being stripped, opens the door to the sea-ward, where he finds the guide ready, and plunges headlong into the water - After having bathed, he re-ascends into the apartment, by the steps which had been shifted for that purpose, and puts on his clothes at his leisure, while the carriage is drawn back again upon the dry land; so that he has nothing further to do, but to open the door, and come down as he went up - Should he be so weak or ill as to require a servant to put off and on his clothes, there is room enough in the apartment for half a dozen people.
In 1901 it became legal for women and men to bathe on the same seashore (one presumes they are clothed) and bathing machines became less popular.
Learn more about bathing and seaside resorts during this era on the following site: