If you’re looking at this from a military standpoint only, one of the first considerations could very well be the National Guard because they are partly tasked with assisting local and state agencies in dealing with emergencies that affect the population of their respective states. It could be the Army or Air National Guard, the Army or Air Force Reserve, or even Marine or Navy personnel that have reserve programs in certain cities and states.
Continuing on, the amount of manpower that an individual unit can contribute to assisting in the emergency comes into play and does that specific unit have service members trained to deal specifically with the crisis that is occurring? (A biological / chemical attack for instance) Whatever the situation may be, all branches of the Armed Forces have teams specially trained to respond, contain, and assist in various types of contingencies. Some branches would probably be better suited than others, depending on what needed to be done.
As far as quicker response times, more manpower, and more resources to draw upon, sometimes active duty bases are better in this regard because some national guard bases don’t operate 7 days a week and their mission requirements may not meet the standards of what will be expected in a crisis situation. A dozen different variables could come into play so there isn’t any easy answers when determining which branch of service would be most favorable, what assets they can offer, and how effective they will be in the crisis management and containment of an incident.