Question:

Does OSHA have any jurisdiction over military installations?

OSHA, and other regulatory agencies, have jurisdiction over "open" installations. If the installation, or any part of it is "closed" to the public, then a different procedure must be followed.
It is routine for access to be dependent upon security classifications, and often outright denied. For example, various parties have tried to use the EPA, OSHA, etc. to access "Area 51," without success.

To confuse things a bit more, even if OSHA gained access, and noted violations they could cite the base, as the employer...however no fines could be imposed. More correctly, fines could be levied, but they would not have to be paid. The military, being a government body, cannot be made to pay a fine by another government body.

DOT is a prime example. Iowa is a state program, so the only inspections performed are state OSHA inspections. If they note a problem at a DOT worksite, they can gripe all they want, but can't fine DOT. Something about taking money from one pocket and putting in another...kinda makes sense. Buddy's a heavy equip operator for Iowa DOT and Iowa OSHA once tried to give a NOV on their site based on shoring of excavations. Didn't fly, but at least the violations were fixed. (Buddy actually called OSHA in.)

Now, were a fed OSHA inspector get peeved at a state DOT site, who knows?

As a former DOD civilian employee... the answer is

When they (fedgov) want them to.

The fedgov routinely applies all sorts of laws and regulation to agencies other than themselves. We called OSHA about a potential violation (asbestos tiles, IIRC) and were told "sorry, can't help you".

Most of what is mentioned above applies to all government installations (we are USDA), though we have found OSHA to be very helpful overall. They do not have the time very often, but when they do, we invite them in. When there is a death they get there (such as a homemade device used to scare off birds killed someone a few years back). They cannot fine us, but I am glad that they can make life miserable for those involved so that we do something to improve. What is better are those sites we invite them to and only minor items are found.