1. "with the" is the correct choice.
That is the city with the worst crime record= That is the city that possesses/owns the worst crime record.
2. "For the" is not correct. "For" implies "purpose" or "result."
The following would not make sense:
That is the city for the worst crime record.= That is the city due to (as a result of) the worst crime record.
That is the city FOR the worst crime record.= That is the city in support of (for the purpose of) the worst crime record.
However, the following would be correct.
That is the *top* city for the worst crime record. (Which means it is ranked #1 due to/ as a result of having the worst crime record)
Hope this helps!