DH is a responsible person and he was that way before we started dating. That doesn't mean I didn't have second thoughts, worries, and a couple anxiety attacks the first times I left our eldest alone with him. Really, it was about leaving my baby at all. That was very hard for me. My imagination went into overdrive and I imagined all sorts of things he would forget, dangers he wouldn't protect her against, etc.
And he didn't always do things the way I would, or as soon. It did take a few conversations and experience on his part to remember to check diapers every couple of hours at most and to realize that she wouldn't always signal hunger clearly and he needed to offer snacks and meals without her prompting. But he got there, just like I did, and now I have no hesitations at all about leaving our children with him.
If your dp doesn't seem to have the instinct, I would start on a small scale, when you're feeling well and there's no emergency, and lay out clear expectations, very concrete things to check for, and really spell it out. He might get offended by you being so detailed, but I do think he needs to understand your concerns and that things like taking a baby outside in the cold without proper clothing simply isn't ok with you (or CPS if a neighbor sees him doing that!). Start small and work up to a level where you and he both feel comfortable. Mamas more often get into the nitty gritty of parenting sooner because they're on, 24-7, from the very beginning. My husband would have loved to have that opportunity, but he had to go back to work within days of both babies' births, so there was a great big chunk of 9-11 hours every day where I got a lot of experience being a solo parent and he didn't, and then he came home and I was there, already in my groove.