I agree with PP. While he's in BCT and AIT, keep the job and use the free child care. He may start getting paid earlier, but worst case scenario plan on not getting paid for his first 1-2 months in the Army. Of course you'll get back pay but military bureaucracy moves slowly.
Wait until after your first PCS (when he gets to his actual permanent duty station) to quit your job because you'll be more financially secure at that point. When you move to his first duty station you're likely to have a lot of out of pocket expenses. Many will be reimbursed but again, that will take some time. You get DLA (dislocation allowance) when you make your first PCS and that will help you pay for the initial cost of curtains, spices/condiments, etc. If the two of you do a DITY (do it yourself) move, you'll pocket some cash from that. And if he's getting a bonus he'll also see that at his first duty station.
Because he's lower enlisted you may also qualify for WIC. Be sure to look into it because that may help you financially and in the long run allow you to be a SAHM.
Before you make this leap, plan out a detailed budget to ensure you can sustain on one income. Your husband needs to maintain good credit in order to keep his job and security clearance.