Mexico is not without some of the blame. Corruption is rife in Mexico throughout the government and police force. This facilitates the drug trade to a degree. However if there was not a demand for it, they would not supply it. The US is the worlds largest drug market. The US should do more to educate and deter drug usage. It needs to be a combined effort between the two governments to fix the problem. No country is solely to blame.
Although Mexico is known for drug trafficking near its US borders, it cannot be forced to take full responsibility for drug trafficking crimes occurring the United States. The United States is responsible for border control. If drug cartel leaders and drugs are smuggled across the border, it is the US's fault that crimes directly related to these leaders/drugs occur.
No, Mexico should not take responsibility for drug trafficking crimes in the US. Each country is responsible for the crimes that take place in their own country. Because the problems do stretch across the border and involve travel, the two countries should work together, but neither is responsible for the other.
Drug trafficking across international borders requires parties on both sides to be actively involved. No one country would ever be able to take full responsibility for such an operation, especially on such a grand scale. Citizens of both the US and Mexico are working in drug organizations, not solely Mexico.
I do not think that Mexico should take responsibility for drug trafficking crimes in the US. Instead, the US government should work on implementing an effective strategy to prevent drug trafficking in the country. I believe that America that find an approach to cut the drug crimes and drug use.