I am looking at this from the perspective of your members.
You are saying this is to discourage membership now that you are at your desired enrollment. Not that your costs have increased or that you now desire to give yourself a raise or your staff a raise. It may irritate your clients if the increase is substantial and it may not do the trick.
Do you have trouble telling people “no”? I used to use my rates as a deterrent to new clients so that I didn’t have to tell people “I don’t want any more clients”. I wanted to keep my free time for me and my family. But this often didn’t work and I felt like I was taking advantage of the people I was charging such a high fee.
I now simply tell people that I can’t fit anymore clients into my schedule and that I can put them on my waiting list for the next available slot. My rates are near the top of my market now and I don’t want to scare away people that need my help so I use a rate range as opposed to a set rate.
Is your problem that you have too many members, or that you think you don’t have enough money? Exclusive or not, tactful communication of any change is going to be important so that you don’t lose more members than you intended.
If you have the right number of members, you just don’t want any more, you might freeze the membership numbers and start taking a waiting list. You only add a new member when an existing member quits. You could also raise your prices for new memberships and keep the existing member rates the same. That keeps your “charter” members happy, and increases your revenue on new entrants.
If you have too many members now, raise the price for your existing members by 10% and then increase the joining fees and monthly rates for new members by 20%.
In your reason for changing pricing, you might focus on how long it’s been since the last time you raised rates and that your costs have increased. You might also mention that the membership increases will go towards something that benefits THEM (not just your profit) like new equipment or another staff person so that they have more individualized attention. Saying “Our gym is exclusive so I’m raising rates to scare people away,” could scare more people away than you intended.