If I make it across the Mexican border into America, can I stay?

No.

Illegally crossing the border, or without inspection, make the person illegal in the US or on that is here illegally. It is unlawful. If and when your caught, you could be deported. You should contact an immigration attorney to determine what legal options you may have to enter the US legally.

If you come without inspection or visa, you are deportable and can be removed.

If you cross the border illegally, you will be subject to deportation. I would recommend trying to find a legal way into the United States.

If you are not caught crossing the border, you will probably stay illegally. If your question is if you can obtain legal papers to stay, the answer will depend on a couple of factors, but Most likely you will not be able to obtain legal papers to stay. Some instances when you will be able to come in and obtain legal papers: examples: 1. If you qualify for an asylum, then you may be able to stay and you should present yourself at the border with sufficient facts to support your case, otherwise, you will be turned back. 2. If you are Cuban and again looking for asylum, you will be let in and you may obtain legal papers. It is impossible to say in your specific instances, without knowing about your circumstances if you can come to the US and stay.

Well, a lot of people do but it is not legal. There are a lot of risks involved in this so please think about finding a way into the US that involves getting a work or training visa or via a family member.