Pretty much what the title asks.
I mean, I know the military can try and punish their own personnel, but what about civilians? Suppose a civilian committed some crime on a military base (robbed the store, say) or against a member of the military (rape, say), would the civilian be tried by the military or by regular civilian courts? Would every crime automatically be a federal offense?
Also, what if the civilian was simply a witness to a crime the military wanted to try a service member for. Could the civilian be compelled to testify? What if he simply didn’t show up for the trial?
There is fairly detailed case law regarding this, and a significant element of it lies in whether access to the normal “Article III” court system is available – martial law during an insurrection, or an overseas base as opposed to one stateside, being examples where it would not be. Perhaps a Doper-at-law or someone with knowledge of the UCMJ could expand on this, as I don’t know the particulars beyond what I said.
In times of a declared war, civilians who are accompanying the armed forces may be tried by court martial under the Uniform Code of Military Justice. Also, subject to a treaty or other international agreement, those who are within a US military base overseas.
UCMJ: Persons who are subject to this chapter
Typically in my day, a civilian getting into trouble on a US base on US territory would be arrested by the military police and turned over to the county judiciary for prosecution.
The UCMJ is (at least on this point) fair-to-middling screwed up. Civilians who accompany the military in the theater of operations (I am going from memory here) during a declared war are liable. Of course we have not had a declared war in about a long time.
The British are not so limited. Their journalists, contractors and other camp followers undergo a ceremony and sign a statement that they (upon entering the theatre of operations) are subject to the Queen’s Regulations.
In a more limited sense, military courts do have power of subpoena and can compel a civilian to testify in the same way any federal court can.
I’d sure like to know what is done if the reverse is the case–a military person–soldier, sailor, airman, Marine–who violates a civilian law, or commits a non-criminal tort, in civilian territory. Are such people under civilian jurisdiction, or military jurisdiction, or both?
(The closest to this I ever heard of was the movie Town Without Pity.) 
Incidentally, something such as is mentioned in the OP happened once, to my knowledge. A serious, seasoned group of bank robbers planned to rob a bank and then escape the police by speeding onto the Florida Turnpike–ignoring the toll booth.
Everything went all right until they got to the “toollbooth” and sped by. It wasn’t a tollbooth. It was a guard shack at the entrance to an Air Force Base. 