Yes, because there is no choice in the matter. The only durable way to stop mass illegal immigration would be to relieve the enormous incentives for both illegals to come in, and businesses to hire them, which means making it legal for a restaurant to hire anybody - including foreigners - to work 16 hour days for $3/hour vs 10 hour days for $9/hour. This will never happen, not even under the most hardcore Tea Party government, not even if Goldwater rose from the ashes and brought with his election victory a House and Senate majority somehow with the same spine he had.
I thought the (excellent) movie "Act of Valor" pointed out the other way this issue could be resolved. In the movie, terrorists use the people smuggling networks to allow suicide bombers into the US, targeting a few stadiums in Southern cities. This is not the first time enemies of the US try to smuggle people in illegally - the Soviets and the Cubans especially sent many agents that way, as well as infiltrating US criminal organisations (see for example Oleg Kalugin's biography [1] which explores the links between the Soviets and the Italian mafia).
In the real world, no criminal organisation would be mad enough to let that happen (much as gangs in Mexico try to avoid killing gringos) because it would mean a promotion from "criminal organisation" to "effective terrorist organisation" and we saw after 9/11 how the American public reacts to that. So long as you're "just" smuggling drugs and people, it's "just" the DEA and ATF and the odd border agent.
If they did facilitate terrorist attacks, you can bet there'd be an enormous appropriation and militarization of the border, coupled with a systematic destruction of virtually all smuggling networks on Mexican soil and maybe temporary occupation of the territory south of the border. Panama was invaded for less.
But this is not going to happen just because a few people go on strike, especially when the majority of manual labour (11m vs 3m) appears to be illegal already without much outcry from the public which tacitly understands that this is key to cheap food and service.
[1] http://www.amazon.com/First-Directorate-Intelligence-Espiona...