husband entered illegally from mexico, procedure for applying for residency

Well, I always recommend a direct consultation directly with a qualified immigration lawyer to see if there is anything unique to your husband's situation which might help. Based on only the facts you present alone, unfortunately I don't see any way.

Generally, you can only get a green card based on marriage to a citizen even if you are here without status - but ONLY if you entered legally, which from what you write it doesn't sound like your husband has done.

Otherwise, the only other way he could get a green card here at this point would be if he were eligible for a penalty provision called 245(i) - a penalty provision that would have let him just pay a $1,000 fine to adjust status in the U.S. even if he entered illegally. but to be eligible, he would have needed to have had some form of green card case filed before April 30, 2001, and that case must have been approvable at the time of filing. not likely since he didn't get here until after that, but worth exploring with a lawyer.

Of course, you could sponsor him and he could go abroad to process - BUT since he has been here more than a year without status, he would be subject to a ten-year bar to coming back. It is in some cases possible to get a waiver of this bar based on hardship to a US citizen spouse (you) or parent - this would need to be more than the normal hardship of losing the presence and support of your spouse, and the possibilities here should be explored with a lawyer.

Good luck!