Yes, President Trump will have the legal authority to ban Muslims from entering the country.
Here is the applicable statute:
8 U.S.C. §1182 (f) Inadmissible aliens - Suspension of entry or imposition of restrictions by President
Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.
You can read about it and watch the video @
Obama ad-Dajjal, on the other hand, does NOT have the legal authority to turn illegal immigrants into citizens. Doing that should disqualify HIM from being President.