One solution to the problem, if the landowner agrees, is to replace a lock with a gate which is bolted shut but requires a large adjustable spanner to undo the nut. This "Derbyshire Key" system has proved its worth for many years in the Peak District.
Most ordinary curious walkers won't carry a large adjustable with them but someone intent on entering the site will know that they can do so relatively easily and will do so without causing any damage. DCA has found that although the nuts are sometimes thrown away, they are easily replaced, so the site can be kept secure and there is no temptation to the vandals because they will know how to get in if they really want to, whereas casual visitors will be kept out. Combine this with a helpful notice and there is normally little trouble. (One of the few sites kept locked in the Peak District has repeatedly had the lock broken and the gate vandalised but, having persuaded the landowner to give the "Derbyshire Key" system a go, there have been no problems since.)
The problem with installing locks which can only be reached through a hole in the door - as OR complains - the vandals may have recourse to an oxy-acetylene cutter to remove the door altogether or cut a chunk out round the lock so that it cannot be locked at all. This actually happened in the Peak District some years ago and, following the wrecking of the gate itself, the site was walled up and there is now no access at all.
If the owner insists on a lock (either combination or with keys) or if the requirement is to protect something fragile within the cave or mine, then you have no choice. However, it might be worth persuading the owner to consider the Big Nut and Bolt arrangement to see if this will work.