What happens if a US citizen gets convicted of a crime before their alien spouse gets their green card?

As a US citizen, you can sponsor an alien for U.S. Green Card application for the the immediate relatives of a U.S. citizen, such as your wife.

But as a convicted criminal, most likely you are in jail, with little or no income. So it is difficult for you to "sponsor" an alien financially. The USCIS Form I-864 - Affidavit of Support is required for family-sponsored Immigration. An affidavit of support is a guarantee to U.S. government that an immigrant will not become a public charge. SO you may need a a co-sponsor for her financial sponsor for Form I-864 if needed.