The answer depends if you're planning to return to the USA or not. It also depends on where you intend to cross. Different bridges have different rules and processes in place.
In my 40 years of experience of crossing into Mexico on foot at the Progreso International Bridge in Texas, I have never once been asked for identification going into Mexico. In fact, last September I walked across (with a large group of hunters) during Covid and re-entered the USA with my Texas drivers license. I usually try to bring my passport but sometimes forget it. A drivers license will work, or you can even use no identification to re-enter, but you will wait for a short interview and some questions. I have done this before after many margaritas.
A green card would not work for entering Mexico, because Mexico does not issue green cards or even use them. A green card is a US form of identification, which holds no bearing in Mexico.
If you intend to travel into Mexico, beyond the border crossing towns and bridges, you will need a passport. A green card will not work with Mexican authorities in the interior parts of Mexico.
If you intend to travel into Mexico, for the day (for shopping, food, medicine, etc) and walk back to the USA, you can use your green card to re-enter the USA. USCIS can easily identify you with it. I hope that helps.