The New Immigration Law & Having A Child In Brazil?

If you overstay a visa (living here illegally) it is R$100 fine per day per person up to the maximum amount of R$10.000

With an anchor baby, please don’t put the burden on the Brazilian tax payer, pay your own expenses.

Aside from the increase in the overstay penalty that Craig mentioned, the new law doesn't change much for a case like yours.
One big change that pre-dates the new law:  Brazil has adhered to the Hague Apostille Convention, so Brazilian Consulates no longer legalize US documents.  You'll need apostilles on all your government-issued documents, from either your State government or the Federal government, depending on which issued the document.
Check the list of documents required on the Federal Police website, and arrive with them already in hand.  The Federal Police will require them to be translated here by a Sworn Translator, so have plenty of money handy for that.
As parents of a Brazilian child, Brazil will extend permanent residency to you and your wife.  It will not allow you to naturalize or issue you passports however - except for your child - unless you establish physical residency.  Brazil does not issue passports of convenience.

We were planning on using a private hospital, so no worries there

I'm only slightly familiar with apostille, does that basically mean that all my documents need to be notorized or does it include more than that?

No, an apostille is a separate, Hague-compliant document, issued by the issuing authority of the original document and attached to it, attesting to its validity.
Because of the federal structure of the United States, a state official,  normally the Secretary of State of each state, issues apostilles for state documents,  and the US Department of State issues them for federal documents.   It's important to send documents to the right authority, or they'll be rejected.
The local office of your Secretary of State may issue apostilles while you wait. If you or your wife were born in the City of New York, there's an extra step, because those records aren't kept centrally with the rest of the state's; the process is explained on the NYS Department of State website.   There's a form on the US Department of State website for submitting federal documents to them.
Apostilles, like the underlying documents, will require sworn translations.

Would I be able to get sworn translations in the States, or would that have to be done in Brazil?

We've got most of the other paperwork and have almost all of it apostilled now. I'm just trying to see what all I can do while still in the States, before we leave.

By the way, thank you. This is hugely helpful!

I'm not aware of any translators in the US with the necessary official Brazilian approvals.  There is, however, a Brazilian firm that will work with you over the Internet,  and provide sworn tranlations with electronic signatures,  that they claim are accepted nationwide.   They are Fidelity Translations (fidelity.com.br).  We had a translation done with them, and they did an excellent job, at 1/3 the price quoted by a local sworn translator; they're willing to negotiate on price.  The document certainly looks official, and our attorney was satisfied with it.   We never had to submit it, though,  so I can't be 100% sure of acceptance.

I checked out the website, it seems like a good possibility. Thanks for the info!