Okay, my Axiom 61-II is stored away at the moment, so I'm using my Yamaha PSR-E443. Also, I'm using Studio One Prime, which is the free version.
(1) You said you set up the Axiom as an "instrument," but that's what you use for sending MIDI to an external destination, typically an instrument or sound module that you want to use to play back the notes that Studio One is sending it.
Instead, you should set up the Axiom as a "keyboard," which is what you use for receiving MIDI from an external source, typically a MIDI controller or external keyboard instrument whose MIDI output you want to capture and record in Studio One.
If you have an actual instrument as opposed to a controller-- for instance, the Yamaha PSR-E443 or similar model-- then you can set it up twice, once as an "instrument" for receiving MIDI from Studio One, and again as a "keyboard" for sending MIDI to Studio One.
But for a controller you want to set it as a "keyboard." You could also set it up as a "control surface" to use it for controlling things in Studio One, but you need it set up as a "keyboard" if you want to use it to play virtual instruments in Studio One.
(2) When you add a new MIDI (or "instrument") track in Studio One, select the Axiom for the input, and be sure to select the virtual instrument you want to use for the output. With Studio One Prime I don't have a choice-- the only virtual instrument available is the Presence soft synth that comes with Studio One.
When you create the new MIDI (instrument) track, you probably won't hear anything when playing the Axiom until after you've selected a program in the virtual instrument. Click on the track's little keyboard icon ("instrument editor") and select one of the virtual instrument's presets, then you should be able to hear something when you play the Axiom-- unless you need to fix your audio setup in Studio One.
(3) If you turn the knobs and so on, you should be able to see the indicator next to the track's keyboard icon go up and down as it registers the activity (incoming MIDI data). If you don't see that, go back and edit the settings for your Axiom in Studio One-- in particular, make sure you haven't checked any of the boxes in the "filter" section, or you'll be filtering out MIDI messages of that type.
If moving the knobs, sliders, etc. makes the indicator go up and down, but doesn't do anything else, you'll need to use the MIDI learn to assign the desired knobs, faders, etc. to the desired functions.
Let me know whether or not any of this helped you to figure out what's causing the problem and to fix it. If it didn't help, let me know so we can proceed further.