A club it was indeed - for men only!
Your snippet says it closed late December 1981, after over a century of supplying gentlemen with an exclusive luxury club. Including a billiard room.
A magnificent 5 floor building on the corner of Warrior Square and the seafront.
From I. Castro, published in The Observer dated June 2017:
Gentlemen’s Club.
This image appeared in ‘The Architect’ in April 1884 with the caption “Gentlemen’s Club, St. Leonards on sea. This building is now in course of erection for the directors of the new club company. It occupies one of the finest sites on the Parade, in the most fashionable part of St. Leonards, and commands magnificent views of the sea coast and Parade. The new club will contain large coffee, smoking and billiard-rooms, and card rooms; also ten bedrooms, bathroom, for the convenience of London and country members. The chief entrance to the building faces Warrior Square, and will be protected from the winds by a large porch, the floor of which, as well as the whole of the corridors, lavatories, are to be laid with marble mosaic by Mr. Burke. The grand staircase’ and entrance framing will be of polished walnut. The building will, in all probability, be lighted by the incandescent electric light, for which the necessary arrangements have been made. The architect to the company is Mr. Henry Ward, AR.I.B.A., Mr. Peter Jenkins is the contractor”. Henry Ward was a prolific and diverse local architect, he designed Hastings & Bexhill Town Halls, The old Observer Building, Debenhams, part of the Buchanan Hospital and many churches in the local area.