There are several ways to send money from the U.S. to Mexico, one of which is to have a relationship with a US bank that is also present in Mexico.
Money being sent between accounts within the same bank is often processed more quickly. Therefore it might be a good idea to open an account locally with the same bank so that the recipient can get access to funds faster.
Transfers between two banks are also pretty easy, but can take much longer with substantially higher costs.
Through banks, you can also initiate wire transfers to Mexico. A wire transfer gets the money to its destination quickly, but there needs to be a corresponding bank to receive the wire transfer. Fees can be very high for international wire transfers, but if the money needs to arrive quickly, this is a viable option.
Yet another way to send money regularly to Mexico is through an ATM or pre-paid debit card. This is not the best method in case of an emergency as it takes time for the card to be sent to the person receiving the funds.
Make sure you have a debit card that will ensure money can be withdrawn from a large network of ATMs in Mexico. If using an ATM card, make sure you trust that the recipient is not accessing funds without your permission. A pre-paid debit card is a safer way to ensure the recipient receives the exact amount of money intended.
Online payment services such as ATM Cash are also a good way to send money to Mexico. These kinds of services allow you to inexpensively send money 24/7 from your own computer.
The above are the most inexpensive, safest ways to send money to Mexico.