I think it depends what you class as "safer". From a long term brain injury point of view, obviously boxing is far more dangerous. (I'll talk more about that in a bit).
Regarding just about every other type of traumatic injury; cuts, bruises, broken bones, strains, sprains, tendon damage etc. MMA is far more dangerous. I boxed really obsessively 3-4 times a week for 5 years and never had an injury other than a hurting my thumbs or a nose-bleed. I've had loads of other injuries in a wide variery of other sports - skiing, snowboarding, surfing, rugby, running etc.
Long term brain injury in boxing:
This is distinct from short term accute injuries like brain bleeds which will kill outright - these are still more common in other sports; like say, Horseriding or skateboarding.
Long term its the repeated sub-concussive blows that take a toll on boxers over a long time.
This is made worse by the fact that boxing is really badly regulated and the medical testing is shit.
Boxers can be banned from fighting in one state, then allowed to fight in another, even though they are known to have "diminishing skills" - symptoms of Dementia Pugilistica.
Any blow to the head will cause a tiny amount of damage and this accumulates over time. However, these risks can be minimised by a unified governing body, with a first set of medical guidelines that all boxers must fullfil to be allowed to fight. If these rules would make boxing more expensive to host then so be it. There'll be less professional boxers and boxing matches.
My personal view is that boxers should be limited to a set number of rounds for their career; say 50x12 rounds coupled with a medical assessment prior to every fight which would assess the cognitive abilities and regular brain scans; maybe every 12 rounds or something.
That would make sure that journeymen don't go on for years and years taking continual blows to the head (although many of these guys are masters of defence) and would also mean that higher level pros who are in hard 12 rounders get more intensive assessment.
I'm equally disappointed by how prevalent drug use is in boxing and how short the bans are. These guys are supposed to be professional athletes. Any drug ban in boxing should be lengthy and really damage a career - at the moment there's not much of a deterant.
I still can't get over the fact that Shane Mosley is regarded as a well respected pro with a great career. The guy is a drugs cheat who was involved in the Balco scandal.