I'm afraid that the novel's publication (1925) predates the 1930s, so I have to disagree with the other answer. I would say the reference is to turning thirty (not "thirties") years old. "Thinning" simply means fewer and fewer single men, less and less enthusiasm, less and less hair on your head, etc.

Hi flying rabbit (love the name), I haven't read the book in a while, but you need to understand the historical context. The 1930s (the thirties) was a time during the 'great depression' (the stock market in America crashed), many workers had no jobs, children had no future. No one was prepared for this because America had stayed out of WW1, but the world, after WW1 was still very unstable. ‎F. Scott Fitzgerald also wrote in a very poetic way, it is why The Great Gatsby is considered a literature classic. "Thirty (1930s)—the promise of a decade of loneliness (a fact that from 1930-1940 people will be alone, isolated. To be lonely could also describe one's emotions; sad, depressed = 'a lonely time'), a thinning list of single men to know (this could mean two things, literally; single men are getting married, no longer single. In a poetic way, it could mean men are dying or losing independence) a thinning brief-case of enthusiasm, (this symbolizes = Enthusiasm is motivation, so a small briefcase would indicate, lack of motivation or none at all vs. a large brief-case of enthusiasm) thinning hair (symbolic; when people get stressed, depressed etc. we may 'pull our hair out' or ''lose our hair')" Ultimately, the sentence reflects the negative and depressing times the 1930s were in America. Hope this helps.