It depends how you define 'climate scientists'. Those that specifically study the climate are climatologists but there are many other branches of science that also involve the study of the climate.
In recent years 18,000 people have qualified as climatologists. It would be reasonable to assume that this constitutes no more than half the total number of climatologists, on that basis the number will be upwards of 36,000. There is no international register of climatologists so it's very hard to provide a specific number.
Some time ago I did read on a skeptics website that the number of climatologists was "only 31,000"
If you widen the definition of 'climate-scientists' to include related disciplines such as meteorologists and paleoclimatologists then the numbers will run into the hundreds of thousands.
Widen the definition further to include astrophysicists, atmospheric chemists, glaciologists and the like and you could probably add another 50,000.
If you were to extend the definition to include all overlapping disciplines such as sedimentologists, hydrologists and dendrochronologists then it's probable that the number of 'climate scientists' would be in the order of a million.