What is the safest way to trigger an output?
Either circuit has failure modes and you'd have to analyze possible abuses such as transients on the 12V line and possible shorts on the output to have some hope of deciding between them. 

Relays are more rugged but have a limited and fairly predictable life. MOSFETs can fail at any time, but in a benign environment, and if minimally stressed (lots of margin on voltage ratings, minimal temperature cycling from self-heating etc.), have few wear-out modes. 

Relays tend to fail 'off' when they wear out (but they can stick) and MOSFETs tend to fail 'on' (but not always). 

If the consequence of failure is an unsafe condition for people or significant property damage, you should not use either circuit alone. Either add redundancy or acceptably mitigate the damage with other means such as fully independent limit controllers etc., preferably using a different physical mechanism to operate. For example, I have a molding machine. The consequence of the platen closing unexpectedly could be a crushed operator. There are three interlocks- one electrical, one hydraulic and one a mechanical rod that physically blocks the clamping action.