Try dragging an Alesis midi file onto the Finale icon and see what you get.

(If you do not own Finale yet, you can download a demo.)

Film arranger and composer Tim Davies discusses importing MIDI info into Finale (by way of a DAW is his suggestion) near the start of his video called "Extreme Australian Orchestration" at http://debreved.com/ 

The issue is the creation of the midi file from the HR16 drum machine to the PC. I do own the latest version of Finale. I think I would need the how to's to set up the midi from the drum machine in a way that the PC can recognize. I'm coming midi out from the machine to midi in of a midisport 2x2 in the B midi in channel. I have my keyboard controller going into the A midi in channel

Although someone here might know the  answer to that, if there is (still) an Alesis user group, that might more readily provide an answer.

Trying to think when I sold mine ... 25 years ago? Technically, I still have an SR-16 but I loaned it to someone about 12 years ago and am fine if I never get it back  The HR is better even if mono instead of stereo  

Heres the manual. Command f should enable search:

https://archive.org/stream/ALESISHR16OwnersManual/ALESIS%20HR-16%20Owners%20Manual_djvu.txt 

In short, there are two sure ways: you can do a sysx dump of all the MIDI data or play the machine into an app that can record and save MIDI data such as nearly any DAW or notation app that supports this function. Finale comes to mind. I’ve never dealt with sysx dumps or importing into an app.