The NSS Cave Files Committee compiles and maintains information on more than 54,000 US caves for use by researchers. That would be one place to start. See
this thread for more info. Of course, the files do not necessarily contain information on every cave in a given state - just the ones that have been mentioned in a publication over the years - but they also include reports from state cave surveys and knowledgeable people regarding the distribution of caves in various states and regions.
Contacting state cave surveys (for the relatively few states where they exist) will be helpful, but not definitive. There aren't that many of them, and some are not very well organized or effective. Others are very protective (some would say secretive) of their information and may not be interested in helping unless you can refine your questsions to specific ones that they can answer (i.e what percentage of the reported cave entrances in your state are on private, state, or federal land?). Getting information from the federal governement can be even more challenging.
Many years ago I was interested in the same questions and spent some time gathering info from the NSS cave files and the state cave surveys that existed at the time. Not surprisingly, I found that the percentage of caves on government versus private land generally mirrored the overall ownership distribution of land in a given state and for the country as a whole. However, I did not consider karst land versus non-karst land in the rough analysis - I just used the overall ownership figures for all land in a state - but I suspect that doing so would not substantially affect the result. It might increase the proportion of publicly owned caves in western states somewhat. Someone else will have to find out for sure.
Generally speaking, in eastern states, something like 95% of all land is privately owned and, naturally, so are 95% of the caves. Out west the ratio of public land to private land is much greater and, correspondingly, the percentage of caves on public land is much higher. I don't recall what it was at the time, something like 50% perhaps, but it should be simple enough to get current figures for public vs. private land ownership for each state. Again, a better analysis would consider the cave-bearing land specifically.
In the end, it was clear that the vast majority of
known US caves are located on private land, which is not very surprising when you consider the demographics of the most cave-rich states - Tennessee, Missouri, Kentucky, Virginia, West Virginia, and Alabama - which account for most of the known caves. Of course, it may be that there are many, many undiscovered (or known but unreported and unrecorded) caves on the vast public lands of the western US. Also, karst is not randomly distributed across the country and state boundaries are political divisions rather than geological, so state to state comparisons (and region to region comparisons) are probably not that meaningful.
If I were considering this question today, I think I would begin by examining the geologic and ownership maps for each state to determine the amount of karst land in public and private ownership. Applying that ratio to the known or estimated number of reported caves in each state, based on the NSS cave files, would give a first-cut estimate of the number publicly and privately owned caves. Further research and discussion with knowledgeable sources could then refine that estimate.