Birth certificates will be fine, but you really should have been considering this sooner. (OFFICIAL Birth Certificates as stated in the link @smokeybandit provided)
Given the long lead times required to get passports right now, you would be lucky to get them in time with only two months until your cruise.
Even with expedited service, it can take up to 10 weeks to process your passport request.
https://travel.state.gov/content/travel/en/passports/how-apply/processing-times.html
If you do feel like leaving the ship in Mexico, it is not necessary to carry passports or birth certificates to get off the ship.
The only time you will need to provide any passports or birth certificates is while checking in on Day 1 and to U.S. Customs at the end of your cruise.
