Gidday,
Firstoff, a baby is usually not a "problem" as such...they grow into that status later in life

so perhaps Congratulations are in order first.
The timing may not be the best for the three of you, but if you're heading off shortly, you'll get most of your initial training done, and may be posted to a new base before the young'un pops out.
I was away for 2/3 of my wife's last pregnancy, and the little guy popped out fine last month. Now I'm at sea again.
There's some good stuff in this thread regarding what you can do for quarters, and such on base.
I'll suggest seeing a doc and getting it confirmed.
I'll suggest that whatever decision you and your significant other make should be done together.
I'll suggest getting ahold of the recruiters and bringng them into the loop.
What follows that is based on what you and she decide, and then what the recruiters are able to help you with. It's not so bad as you think. I don't know what the recruits earn nowadays, but I know for sure they earn a lot more than the $1225 a month I got when I was in basic. Once you're done basic, you're making the better part of $30K or so....a liveable wage.
So, congrats to you, and I hope your dilemna isn't keeping you up too many nights. The baby will later
NavyShooter