Generally yes, asylum should not be a catch all for vague concerns over potential violence. The simple fact is, Mexico and much of Central America is a shit hole. Nearly every single person there lives in poverty compared to the American standard of living. Should we allow EVERYONE to come in simply because their country is a shitty place? That's ridiculous.
By simply following the logic you must either give everyone who wants it asylum or you have to restrict to specific provable concerns that the powers of a country are persecuting people. The reality is that much of the immigration is economic in nature NOT fear of government or even cartel reprisal.