The first expense is of course rent and taxes for the land the spaceport is on, after that there are a number of infrastructure maintenance expenses involved. A basic spaceport is going to have a lot to maintain. For example the Kodiak complex has a clean room for final satellite preparations, a vehicle assembly building, as well as a mission control center with communications and computer systems. Large buildings like these, especially ones with complicated HVAC systems, like clean rooms, and IT infrastructure, like mission control, will require constant maintenance. Kodiak seems to only launch solid fuel rockets, but other launch sites use cryogenic liquid fuel or oxidizers, requiring extensive tank farms with added maintenance requirements. On top of that the pads and roadways will require upkeep as well as other general groundskeeper like tasks (cutting the grass or shoveling the snow in Alaska).
For a small site this could likely be done with a smallish staff likely <50 people onsite full time, this number would go up with a launch bringing in a lot of mission specialists and other personnel. There are also going to be the normal materials and equipment costs associated with maintaining the infrastructure. A big cost is transportation, these launch sites are usually in isolated areas making moving people and materials there expensive. A large part of the expenses are also offsite, from the corporation running it, which has business offices in Anchorage and Huntsville with staff in each office to pay for, normally these would be minor expenses, but it all adds up when you only average 1 launch a year.
Their operating expenses are only around 10-15 million dollars a year.
Kodiak has a lot of information on their website, including annual reports with detailed breakdowns of costs and revenue.