Spoiler: Probably not.
Last week, President Obama ended the "wet foot-dry foot" policy for Cuban illegal aliens. Under the old approach, any Cubans apprehended at sea by the Coast Guard were returned to the island (unless they made a credible case for asylum), while those who reached shore were allowed to stay, enabling them to get green cards after one year, per the Cuban Adjustment Act of 1966.
Wet foot-dry foot is itself a narrowing of earlier practice; before President Clinton formulated it in 1995, all Cubans, even those at sea, were simply brought to the U.S.
President Obama's announcement last week is actually within his authority (for a change). The Cuban Adjustment Act gives automatic status only to those Cubans who are paroled into the United States – but if a Cuban illegal is detained and not paroled in, the Act doesn't apply. And whether or not to parole them in is up to the executive (though you won’t be surprised that Obama has abused this narrow power in a variety of other ways).
Under the new policy, Cuban illegal aliens – whether rafters washing up in Miami or the far, far more numerous "dusty foot" illegals coming through Mexico – will be treated like everyone else. They will be considered for asylum if they request it, but if not (or if they have no credible case), they either won’t be let in at all or will be kept in detention pending the resolution of their case.
This change is long overdue. As I noted on these pages exactly a year ago, most Cuban arrivals are now just regular migrants, coming for the usual mix of reasons – jobs, family, and welfare. (The Ft. Lauderdale paper's blockbuster series on widespread welfare abuse by newly arrived Cubans prompted legislation from Sen. Rubio and Rep. Curbelo.)
Obama's move was prompted by the normalization of relations with the island in 2014, which fueled a surge of new illegal immigration trying to beat the very change announced last week. Of course, the normalization of relations should have been contingent on making the immigration change from the start, but this kind of prioritizing of diplomacy over the safety of Americans isn’t unique to Obama; Clinton established diplomatic relations with Vietnam over 20 years ago without that country first agreeing to take back its own citizens we wanted to deport. (As of six months ago, there were nearly 8,000 Vietnamese-citizen criminal aliens with removal orders whom Vietnam won't take back.)
As to the question in the headline – will President Trump go back to the special policy for Cubans? I think it's unlikely. As President Obama just said at his press conference, our Cuba-specific immigration policy was an anachronism. The angry commentary in the wake of Obama's announcement (see here, for just one example) is obviously correct that Cuba is still a dictatorship. But the Soviet Empire of which it was once a part has been gone for a quarter-century, and Cuba is now just another decrepit Third World country ruled by gangsters, no different from Zimbabwe or Laos or Turkmenistan or Russia. People from those countries do not, and should not, have automatic, unlimited access to the United States, and neither should Cubans.
More than 56,000 visa-less Cubans were waived through U.S. ports of entry last year under the old policy, up 30 percent from the prior year. And that's only the ones coming by land through Central America and Mexico; a smaller, but still significant, number came by sea. I don't see a newly sworn President Trump restarting that large flow of illegal aliens. Such reinstatement is doubly unlikely because the people you’d expect to champion it are not doing so; if Marco Rubio won't ask for the old policy to be restored, who will? (Though he did say he'll press the new president to restore a smaller program that facilitated the defection of Cuban doctors stationed overseas, which Obama also ended last week.)
Of course, this means Cubans will now join other prospective illegal aliens in trying to sneak across the border, rather than just walking up to our immigration inspectors and demanding to be let in. Better get that wall built . . .