In the period leading up to the Bell Riots, their budget was cut several times, including shortly before the riots, making them overworked and underpaid. When the riots broke out, they quickly lost control, and the situation was resolved by SWAT instead.
They were only mentioned as named "district Security" in dialogue, but the term "Sanctuary District Police" can be seen in signage outside the main gate to the District, and this is presumably their official name.
The District Police might be operating under the San Francisco Police Department, though their uniforms seem distinct. A helicopter was heard over the district during the riots, it's unclear if it belonged to District Security, the SFPD or even the state National Guard which was also present.