(Assuming there were grounds for arrest and prosecution).
Nobody knows for sure. The Constitution gives the House and Senate the (sole) power to impeach and remove a president, but that does not preclude a President being prosecuted for a crime. The Constitution does not distinguish the between the president and other high officials in terms of alleged immunity: it says
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors
which includes federal judges. Harry Claiborne was a federal judge who was criminally convicted while holding office, and was impeached after imprisonment because he threatened to return to the bench.
In US v. Nixon, the court found that
Neither the doctrine of separation of powers nor the generalized need for confidentiality of high-level communications, without more, can sustain an absolute, unqualified Presidential privilege of immunity from judicial process under all circumstances
That case, though, was about subpoenaing records, not prosecuting a sitting president for a crime (it was ruled that he had to comply with the subpoena).
It is reported in a NY Times article that the best arguments on the topic are contained in the briefs in the Nixon case files by St. Clair (for Nixon) and Jaworski (special prosecutor). SCOTUS avoided making a ruling on that issue.
Presidential privilege has generally been assumed to extent only to the official acts of the President while in office, which is why President Clinton was allowed to be deposed related to unofficial conduct forming the basis of a civil lawsuit.
So, if a sitting President, for example, beats up his wife while in New York State, he could (in principle) be arrested and prosecuted for that crime while in office, although he would likely be given every courtesy and benefit of the doubt in terms of pre-trial release, etc. while the case was pending.
Similarly, Governors of U.S. states have been divorced and subjected to court orders related to their divorces while in office in the courts of their states, and there is no reason to think that a U.S. President couldn't be subject to the jurisdiction of a divorce court if he or his wife sought a divorce while he was in office, and presumably, he could also be subject to either civil or criminal contempt if he violated a court order in the divorce case, at least if the violation was not intimately related to his official conduct (e.g. the divorce court couldn't order him to take actions setting U.S. policy on some matter).
The Supremacy Clause would probably bar any prosecution of a sitting U.S. President by state or local officials for any conduct colorably related to his official conduct while in office (e.g. allegations of war crimes in how the war in Afghanistan was conducted that were allegedly made upon his orders).
Presidential privilege is quite a bit more expansive in the case of a President than, for example, a judge or prosecuting attorney, because the scope of what portion of a President's activities constitute his official actions is much broader than the scope of what constitutes most other officials' official actions. For example, if the President has dinner for a foreign diplomat, he is still carrying out his job.