A story in Newsweek asks that very question.. Some say it might:
Many political analysts compare the current tumult over the Arizona law to the uproar over Proposition 187 in California, which passed with GOP backing in 1994 and sought to bar illegal immigrants from receiving public services (it was later declared unconstitutional). In the four years after the measure’s passage, California Hispanics registered to vote in droves and chose the Democratic Party over the Republican Party by a ratio of 8–1, says Harry Pachon of the Tomás Rivera Policy Institute. That helped turn California, the largest state in the country, where Ronald Reagan and Richard Nixon hailed from, into a reliable Democratic bulwark. Analysts also draw parallels between the Arizona law and the “Sensenbrenner bill”—named for its Republican sponsor, U.S. Rep. James Sensenbrenner—that passed the House in 2005 and could have led to massive deportations of illegal immigrants (it never became law). That proposal triggered huge demonstrations in 2006 and helped Democrats win sizable Latino margins in the midterms later that year.
Others say maybe not:
In a poll taken prior to the passage of the Arizona law, a survey by Latino Decisions found Hispanic political engagement “at an all-time low,” according to Gary Segura, a member of the polling firm. Only 49 percent of Hispanic registered voters were very enthusiastic about voting in the coming November elections, compared with 89 percent in September 2006, during the last midterm cycle. (A February survey of four states by the Joint Center for Political and Economic Studies, on the other hand, showed that between 74 and 80 percent of African-Americans said they were very likely to vote in November.)