1) Did you even know that Basil was a man’s name, or do you just think of it as a garden herb that’s good with tomatoes. I would think of this
2) Do you go straight to thoughts of Basil Fawlty and hear Sybil calling Basil! and this
3) Could YOU name a child Basil? Really? and to be honest no
If you like it then go for it. I like the nickname. As they say when naming your kids. Say the name out loud a few times.. ie.. Basil/Baz pick up your toys.. Come here Basil/Baz.. Think of how it might be for him later.. If you still really like the name but aren't sure then maybe use it for a middle name