While I think that military members should be able to carry on base I see some issues (stated in no particular order):
1) Weapons Familiarity/Mindset - At least in the AF, there isn't much focus guns. In fact unless you deploy regularly, it isn't uncommon to not even touch a gun except for once every 1-2 years for qualification. Hell even at my deployed location I have a lot of people who question why we even have our guns since "we will never need them." At home on my last airframe I was viewed as "odd" for desiring to carry a weapon at all (either on or off duty). Then there's the curiousity aspect for those that would want to carry but don't really have any experience with weapons (the "little kid" syndrome as someone else described).
As such I think that both the weapons familiarity and the mindset regarding weapons in the military (and especially the AF) would need to change in order for the masses to safely carry.
2) Gangs and those that snap - The Army and Marines have gang problems (the other services too, but not to the same extent) as gangs have started sending members to the military for training. Also people are more likely to "snap" at home station. Both of these people present problems when armed at home station. The criminals are to potentially feel empowered due to their gun (and either intimidate or actually use it) and some of those that snap are likely to decide to use their weapon on those they blame.
The military needs to better get the gang problem under control and better support those that might snap (PTSD, excessive stress, etc).
3) "Respect" - I would say that too many officers DEMAND respect without actually doing anything to earn it. The gun can make subordinates feel empowered and thus more likely to challenge such less-than-stellar officers. So I can see certain officers not wanting this as it would require them to actually earn their respect instead of simply demanding it.
4) Carrying of weapons when offbase but in uniform - While I'm not against one carrying their own weapon so long as it is equal or better than what the military would issue AND conforms to certain standards (such as compatible with a specific mag and ammo-type), there would be the question of what happens when one leaves the base but is in uniform. We're NOT police (even military police aren't authorized to perform police actions off-base unless its involving another military member), and seeing a bunch of military walking around in uniform with weapons can potentially scare the general populous. Not to mention that invariably someone would do something they shouldn't with their weapon. As such it is a potential PR problem for leadership.
Note that I'm NOT saying that due to these reasons we shouldn't be allowed to carry, but rather that these are the reasons I can see our "leadership" not wanting us to carry. Not to mention that I'm sure plenty are anti-gun as shown by General McChrystal's
comments after one of the mass shootings. Personally I think the biggest thing would be promoting a safety mindset, breaking down the curiousity that many would have, having proper training that ANY military member who desires to carry can take, and having serious consequences for those that act improperly with their weapons (like loss of multiple ranks, jail time, CC, AND wage garnishment), but I just don't view enough of our leadership as truly pro-Rights and so I don't see them having any desire to enact something like this.