Well, there is a lot of (sometimes badly written) Russian Sci-Fi on "how to fix past things with time travel", where most involve "not being caught by KGB" (or what was its name in the era). I am borrowing heavily from those.
Use (older) spy techniques
Record the data is some way, stash it somewhere safe in the past (you have the time, since no one knows you are stashing something), alert your government contact / journalist of your choice / local police officer.
If you want, the alert step might involve BBS, but should not. The good thing: the actual live communication is short, less options to be traced.
As far, as I understand USA, you can just hook a modem to a phone in a phone booth in the middle of nowhere, send a short message, unhook the modem, and drive away.
Further sources of inspirations would be the communications operations by some intelligence services done before small electronics kicked in. Read the memoirs! Or some mafia transactions of the kind "let's exchange this suitcase with money for this suitcase with cocaine and let nobody get shot".
Since your protagonist is stashing a memo (about 9/11, Kennedy assassination, etc.), they might just write it down. As long as no one is there to catch them and to demand a comparison of handwriting in court, this is Ok. Extra bonus for some too good or unusual typesetting. (For example, folks from 1930s would be rather surprised to see modern bleach-white paper printed with a laser printer. They will be able to find out, this is not printed in their-modern way, trust me. Or just take a look on differently-printed paper under a microscope.) If the memo is openly stating "I came from the future to prevent XY, this adds credibility.
You might use some data storage, you know, exists in the era. Such as a set of floppy disks or microfiche.
The best approach I have seen, was to:
- Use stenography for a compact writing (requires learning it, but helps a lot. You can also use computer-based typesetting, there were some approaches for that.)
- Fake foreign handwriting when doing so (hard, might be unrealistic, so this is optional)
- Photograph the written sheets on a usual film camera (requires some fiddling, but is generally doable)
- Place the film in the hideout from above
- Alert the KGB chef, his package lies in the secluded place.
Basically, your recipient would be responsible for picking the film up, developing it, printing the frames, decoding the writing, acting on US missiles in Turkey (or what not). Guess, what is the hardest part?
Again, if a computer might play a role, then a (modern) one in 1 and (contemporary) one in 5, with a modem and somewhat secure / pseudonymous connection.
You do not want the session to be fully anonymous, it suffices not to be caught. Hence, "random entry points" cross the country might help. Or hacking. Oh, the hacking!
In the 60s, there was a lot of action in the telephone department. Some custom-made devices could play the role of the phone company equipment and do magnificent things. Of course, they were illegal. For example, a blue box allowed you to call long-distance for free.
Using such a device for the transmission would not throw a large, state-sponsored intelligence division completely off-track, but would help to slow them down. There was/is a strong community around the colorful boxes, so some connections and a lot of googling in their time would help the protagonist.