My partner is afraid of heights, and of needles. He can't even watch if someone is getting a needle on TV. He can get vaccinated, though he dislikes it, and he's been lucky enough not to need a lot of IVs or blood samples. His fear of heights extends to not being ok with me looking over a lookout railing or going near the edge of a cliff. This has spoiled a couple of hikes over the decades.
Both of these have improved dramatically in the last 5 years, rather accidentally. We took an amazing trip which included many hikes that went along the edges of cliffs with no guardrails, something we hadn't expected. (These countries have different approaches to guardrails than North America.) There was no choice but to walk along the high paths for some time, and he now is actually much more comfortable with some heights, though he isn't going to seek them out. Also, I got very ill and have been having IV treatments, blood draws, surgeries etc for about 18 months now. For most of these he has had to accompany me and be helpful, such as knowing when to squeeze my hand and when to let go. This level of exposure has also reduced the fear for him.
If your partner genuinely wants to donate blood, then it would be ok for the two of you canto try together to work your way up to it. (Don't do this if it's you who wants him to donate blood or to get over the fear.) The first step, which you're doing, is to come with you and see all the aspects of it except the donation part. The second would be to come in with you, but turn away for the needle part. He'll get the sounds and all the other experiences, and can see you there with the needle in your arm not having a problem. After a few more times, he might watch the actual stick (but he doesn't have to, I don't watch my needles go in. I do keep my eyes open though because that makes the needle go in better, I don't know why.) Eventually he might be willing to give it a try. It would help if over the time you were working up to it, the pair of you got to know some of the blood techs, so you could be sure of a compassionate and pleasant first attempt. Trust me, they vary wildly and some are jabby and curt while others work with you to make it go smoothly.
In general, people's fears are their own, and for them to face or live by. I don't want someone "training" me to be willing to park in underground parking or other things that scare me. If your partner's fears are holding you back, working together on his behaviours is fine. And if he expresses a wish to do something, you can offer to help. But curing his fear is not your task; don't take it on.