Assuming we are operating on our own and not with allies which NEVER happens....
Immediate (aka: 5 min ago) AF security you would send someone from CANSOFCOM. JTF probably would provide their own security for their own birds to start, or CSOR would be involved. JTF often provide their own security as they traditionally don't trust anyone else. It all depends on the mission and where.
For a week or so notice any of the Infantry Btn's can fill the shoes.
For longer term with more lead time or multiple rotations you could issue a class C. Plenty of NAVRES sailors provided base security in KAF and with the SOF units (as I was told they didn't have to "relearn" anything that was taught them by the SOF guys, blank slates and all, that unlike combat arms PRES), any combat arms PRES units like in Camp Nathan Smith.
Class C's for these sort of tours that get you a medal are gold to PRES. They love them, get a good paycheck, break from school, work, life etc...and get a shinny to show off in the mess. For the platoon sized org that is required for airbase security you could probably get Class C's for an infinite amount of time. During the heydays of Afghanistan it never seemed like we were ever short of reservists volunteering to help out.
We can't do AAW defence for airfields aside from MANPADS AFAIK anymore.