Single story 1960 home, 1260SF in California climate zone 3C (95062). 

The house is currently uninsulated and we’d of course like to change that. I am concerned (perhaps overly) about insulating the home and causing moisture problems. Our sheathing appears to be a 5/8″ gypsum board product. The ‘paper’ is black so I’m assuming this is some type of asphalt felt paper. Outside of this, we have metal wire and stucco. 

I’m planning on using mineral wool. In my head at least, the hydrophobic properties of mineral wool would alleviate some of the moisture issues if there were any. I’m not sure if there is any truth in that, however.

Does this seem reasonably safe to simply throw batts into? Anything additional I can do like caulking the seams of the gypsum board panels?

Pictures attached.