There are a few reasons I have safety concerns about Tiger Moths being used to carry members of the public:
1) fuel tank above an open cockpit, also any overflow from the fuel tank is likely to spill into the cockpit.
I'm not one to believe Tigers are inherently unsafe, but I know of one incident in a Tiger where this was a factor.
While refuelling the aircraft, a small amount of fuel had spilled onto the front seat. The fuel soaks into the front passenger's pants some time into the flight, and when he eventually feels the associated burning sensation around his groin he starts screaming that the aircraft is on fire.
In response, the pilot made a forced landing in a paddock. Both persons on board walked away, and there was no fire, but the aircraft suffered substantial damage.